There are many points during the process of applying for a U.S. visa or green card at which applicants might be tempted to lie. For example, they might think that it would be harmless, or essential, to hide a ground of inadmissibility, ignore a previous marriage, or avoid questions about previous overstays at the end of visits to the United States.
But lying to the U.S. government can get you in bigger trouble than the problem you are lying about.
Credibility is hugely important in any immigration application. Some immigration applications literally rest on your credibility; that is, whether the immigration officers feel they can believe you.
Let's say, for example, that you came to the U.S. as a student, and are applying to adjust status (get a green card) on the basis of your recent marriage to a U.S. citizen. But you had a little trouble finding your divorce certificate, so you just left out any mention of your previous marriage on your adjustment application. This misrepresentation is discovered, and you are denied the green card and placed in deportation (removal) proceedings. Your new spouse is angry, and divorces you. In court, you tell the judge, "But I can't return to my home country, I'm a member of a persecuted minority, and might be killed." Ordinarily, that would be grounds to apply for asylum, which could save you from deportation. But the judge might very well deny your asylum application on credibility grounds. If you've lied once, the judge would have solid reason to believe that you might lie again, and be making up your fear of persecution.
In any case, if a USCIS or State Department official discovers that you've lied to him or her, it casts everything else you've submitted into doubt. Before approving you for anything, the officials must be convinced that you're not concocting a fraud or scam.
Making misrepresentations on an application for immigration benefits is a ground of inadmissibility under U.S. immigration law. (I.N.A. § 212(a)(6)(C)(i).) So if the lie is discovered, you've got double trouble: not only the original problem about which you thought it necessary to lie, but a new bar to your being admitted to the U.S. in any capacity. (See Inadmissibility: When the U.S. Can Keep You Out for more on this topic.)
If you feel you just can't complete an immigration form, or attend an interview, without hiding a certain piece of information, or you really don't know how to answer or explain a key question, see a lawyer. The lawyer can analyze whether you really have a significant problem, and might be able to show you how to be truthful in a way that doesn't risk having your application denied. The lawyer will also know how to make use of any exceptions that may apply in your situation, or whether there is a waiver (legal forgiveness) that you can apply for.