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A. Means-End Analysis
Here is the simplest and most accurate way to think of the various levels of scrutiny when applied to a discriminatory law. Each of the levels of scrutiny is basically a means-end analysis.  
Let��s look at strict scrutiny first, and then the other levels of scrutiny will be more easily understood in comparison.  In the case of strict scrutiny, the question is whether the discriminatory law is a highly efficient and effective means to some extremely important end. Or, stated differently, but also saying the same thing: is the discrimination in question a necessary, carefully written means of achieving some super-important goal of the government.  So in a case applying strict scrutiny, each side in a lawsuit, both the government and the person discriminated against, will argue whether there is some extremely overriding (compelling) government goal/end (interest), and is the law a carefully chosen, close and direct means (narrowly tailored) of achieving (furthering) that goal/end (interest).
It��s important to understand that the person being discriminated against can NOT simply claim ��this is discrimination�� or that ��my rights are being violated.�� The first statement is given in the facts; the second statement is the issue to be decided. Laws discriminate all the time. Some discrimination is constitutionally permissible, some of it is not. In other words, sometimes discrimination violates rights, but often it doesn��t. Laws that prevent individuals under the age of 18 from voting, or that deny food stamps to the wealthy, or that deny admission to a university unless you have a high school diploma all discriminate, but all are perfectly legal forms of discrimination under the Constitution.  So the relevant legal question is NOT whether there is discrimination, but whether there is arbitrary or irrational or unreasonable discrimination.
Strict scrutiny is the legal standard for determining whether any particular example of discrimination based on race, ethnicity, or national origin, or any type of discrimination involving a constitutional right such as marriage, travel, voting, free speech, religion, etc. is legally/constitutionally justified or legally/constitutionally unjustified. In other words, is that particular type of discrimination in that particular context or instance legally justified or legally unjustified.  Intermediate scrutiny, then, is the legal standard for determining whether any particular example of sex discrimination is constitutionally permissible.  Minimal scrutiny—also called the rational basis test—is the legal standard for determining whether all other types of discrimination are constitutionally allowed.   So both sides need to focus their arguments only on the precise elements of whichever legal standard applies, to determine whether the discrimination in question is legally justified or legally unjustified.
On a related note, it��s important to remember that none of the levels of scrutiny looks at the interest of the person being discriminated against! That is to say, strict scrutiny, intermediate scrutiny, and minimal scrutiny each look only at the means-end test described above, and do NOT weigh or balance any competing interests of the government and the individual against each other. All that is legally relevant to the analysis is the government goal/end (interest/objective) and the government��s chosen means to achieve that goal.  So, for example, in the case of strict scrutiny, the government will argue that they have a super-important goal, and their chosen discriminatory means directly achieves that goal. The individual being discriminated against will argue the government��s goal is nothing important at all, and even if it is, the government��s chosen means is a highly inefficient and ineffective (or even counterproductive) way of achieving that alleged goal.  The possible motives, intensity of feelings, or claimed interests of the person being discriminated against are all irrelevant under this analysis.  
A. The End
Looking at the government interest involved—the legislature��s alleged goal or end behind the law--is usually the easier part of the means-end analysis.  In the case of strict scrutiny, courts have found several different ��compelling�� interests on the part of government. Among these are health or safety concerns where death or severe illness/injury is possible, national security, military urgency, preserving the democratic process, protecting constitutional rights, preventing constitutional violations, and achieving racial equality in society. Relatively few discriminatory laws can meet this level of scrutiny.  ��Important�� government objectives under intermediate scrutiny is a broader category of possible government interests than the category of ��compelling�� interests.  Thus, more discriminatory laws can meet this standard than can meet strict scrutiny, but still only a minority of discriminatory laws manage to satisfy this level of scrutiny when it is applied.  ��Legitimate�� or ��valid�� government interests under minimal scrutiny is an extremely broad category of possible government interests, as every law is going to be based on some alleged reason.  There are as many reasons for laws as there are laws on the books—which is theoretically infinite.  The only question under minimal scrutiny is if any given law has a rational basis for its existence.  If so, the law is constitutionally permissible. If not, the law is unconstitutional.  A law would have to be completely arbitrary or irrational or unreasonable—that is, the law would have to have no conceivable rational basis---in order to fail this level of scrutiny.  Thus, the vast majority of all laws can satisfy this minimal level of scrutiny.
B. The Means
Understanding how to analyze the government��s chosen means is generally more difficult than understanding the reason motivating the law.  Let��s begin with strict scrutiny and its requirement that the law be ��narrowly tailored to further a compelling government interest.�� What does ��narrowly tailored�� mean? Think of a suit or a dress: its goal (interest) is in covering your body.  But then what is a ��tailored�� suit?  That means the suit has been custom fitted to your own particular body. A ��narrowly tailored�� suit or dress is fitted so that it fits close to the shape of your body with no excess in fabric used—it is a trim fit, not a baggy fit.  There is no unnecessary material that has been used to craft the suit.  Similarly, a ��narrowly tailored�� law is carefully fitted to the goal the government is trying to achieve, and uses no more ��fabric�� (no more discrimination) than is absolutely necessary to meet the goal that the government wants to achieve.
To return to our clothing analogy, if a suit or dress is not narrowly tailored, the suit or dress would look baggy on you because it uses more fabric than is necessary to cover your body. Think of a clothing shop that gives you a suit that hangs off your shoulders, is several chest sizes too big, the waist goes all the way down to your knees, and has sleeves that go down past your hands. That suit is NOT narrowly tailored—not even close! It is way too big, and covers way more of you than is necessary to meet the goal of covering your body. A shopper who wants to prove to the clothing store that the suit is not narrowly tailored would show the clothing store a better-fitting suit, and how it uses much less fabric to still cover your body while fitting better at the same time. The better-fitting alternative shows that the original suit you were given is NOT narrowly tailored.
Similarly, if a law is not narrowly tailored, it uses more discrimination than is necessary as a means of ��fitting�� the law to the government��s goal (some compelling interest in health, safety, national security, racial equality, etc.). The person who is being discriminated against and is challenging the law will thus argue that the law is not narrowly tailored to achieve the government��s goal. That is, the person will argue that discrimination is not a necessary means of achieving what the government is trying to achieve—the discrimination is unnecessary because some other law that does not discriminate would still achieve the government interest at stake. To phrase this using our clothing analogy, the person challenging the law will argue that the discriminatory part of the law is excess fabric/baggage that could easily be trimmed/removed/cut out of the law because if the law were carefully rewritten in way that does not discriminate, the law would still cover all of the ��body�� of the goal (interest) the government wants to achieve. The law would then be "narrowly tailored" to fit the goal.  The simplest and best way to show that a law is not narrowly tailored is by coming up with a specific non-discriminatory alternative law that still achieves the specific goal the government seeks to achieve.  By providing a concrete non-discriminatory alternative that still achieves the government��s interest, the person challenging the discriminatory law thus shows that the discrimination could be ��cut out of�� the law (like trimming all the excess fabric off of the baggy suit) yet still achieve the government��s goal.  Thus, the challenger tries to show that the law is not narrowly tailored. The government, on the other hand, will argue that unfortunately there is no other way of achieving its goal but to treat people differently; discrimination in this case is the best and only way to achieve the government��s end.   There is nothing the government can cut out of the law; it has already trimmed the law as much as it can to fit closely to the government��s goal, and even with the discrimination involved, the law is narrowly tailored.
To continue comparing the means-ends analysis of levels of scrutiny with the analogy of a tailored suit, recall the question is how well the means fit the end.  Imagine that strict, intermediate, and minimal scrutiny were applied to the design of professional suits, with the ��interest�� at stake being covering one��s body.  In strict scrutiny, no excess fabric is allowed; the suit must be trimmed so that it very closely fits by being ��narrowly tailored�� to your body. In intermediate scrutiny, the suit need not be ��narrowly tailored,�� but must still "substantially�� relate to or further or promote the interest of covering your body.  Thus, a little excess fabric in the suit will be allowed.  The suit can be slightly baggy, but not too baggy.  It still must be a fairly close, decent fit, but not necessarily a perfect fit.  In minimal scrutiny, the suit can be very, very baggy, as long its covering of your body is ��conceivably rational.��  Thus, it can have lots of excess, unnecessary fabric that is used, as long as it still meets the goal of covering your body, even if it does so in an obviously sloppy way.
Finally, let��s looking at some visual examples of this analogy below.  The suit on the left would be good enough to pass intermediate scrutiny, but it��s not quite narrowly tailored as it��s a little baggy in a few places.  The suit on the right is narrowly tailored, though, as there��s no more fabric you could cut out yet still have the suit cover your body in a way that fits your particular body.  The third suit below would probably fail both intermediate and strict scrutiny, as it doesn��t even come close to being a good fit on your body.  That suit would pass minimal scrutiny, however, because it does at least work somewhat towards the goal of covering your body.  Not in any exacting way, obviously, but at least somewhat, which is all that minimal scrutiny requires.  

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