I doubt you’ll be able to find sample tests for free, since there’s such a market for them, and anyone with sample tests will want to make money. There may also be copyright issues.
But I can describe them. There are two categories, verbal and math. On test day, you usually take five or six sections (I may be misremembering on this part, it’s been a while), two each on verbal and math which count for your score, and another verbal or math section which doesn’t count for your score, but is used for statistical purposes (you don’t know when you take it which section this is, so you have to put in full effort on all of them). Each section has a time limit of (I think) 30 minutes. Within a section, you can do the questions in any order you wish, skipping over some and possibly coming back to them later, but once the time is up, you can’t go back to any of the questions on that section.
The format is multiple choice. I think they’ve added some essay questions, too, but that was after my time (thank Heavens). You have a newsprint booklet which contains all of the questions and five answer choices each, and you record your answers on a separate bubble-sheet with a number 2 pencil. A calculator and (I think) a straightedge are allowed for the math part, though there are restrictions on what sort of calculator you can use (when I took it, you weren’t allowed a graphing or programmable calculator, but I think the requirement is more lax now). Your score is based on the number of questions you get right, minus a quarter of the number you get wrong, so on average, random guessing won’t improve your score. So if you have no clue at all on a question, leave it blank. But if you can eliminate one wrong answer, or have a hunch, then it’s to your advantage to guess. Your raw score for each subject (math and verbal) is converted to a number ranging from 200 to 800, incremented by 10, by an arcane and incomprehensible formula. The formula is supposedly designed in such a way that scores from one year mean the same thing as scores from any other year (that’s where those statistical sections come in). Nobody knows why the scores are 200 to 800, other than just that that’s the way it’s always been done.
The subject matter is fairly straightforward. The math questions don’t go any higher than what they call trigonometry, but you need never touch a sine or cosine. All you need know is the Pythagorean Theorem and that the sum of angles of a triangle is 180 degrees. There are also some basic algebra problems (2x + 3 = 7; what is x?), and some basic logic problems. In the verbal section, there are reading comprehension questions (you read a few paragraph passage, and answer questions based on it), analogy questions (cat is to fur as snake is to ____), and lots of vocabulary questions (a synonym for “pulchritudinous” is ____). You can take the test as many times as you like (and can afford); generally, only your best score will count.
There are a few other similar tests. The PSAT is taken earlier in high school, and serves primarily as practice for the SAT, and also as the first step in the National Merit Scholarship Program. It’s identical to the SAT, except that the scores for each subject are 20 to 80 instead of 200 to 800.
And the ACT is another test which serves the same purpose as the SAT: Some colleges require one or the other, some require both, and some give you a choice of which one to use. The ACT is divided into five subjects, rather than three (science, math, and reading are three of them; I don’t recall the other two), and the scoring for each subject is from (I think) 1 to 32. Like the SAT, the ACT is primarily or totally multiple choice, but unlike the SAT, there is no penalty for guessing. Your score is based solely on the number of questions you get right, and a wrong answer counts exactly the same as an answer not filled in. So when you get to about a minute left, you should just go through and fill in “c” on every question you don’t know (or fill in “a”, or make pretty patterns in the bubbles, or whatever).
Finally, there is the SAT II, made by the same people who make the SAT, which is a collection of different tests on specific subjects. There’s one for Latin, for instance, and one for physics. These usually aren’t required to get into the college, but they may be required for the major you’re going into. Typically, a good score on such a test may let you get out of one of your first-year courses. These are primarily multiple choice, but when I took them, they all had essay sections as well.