How do pools work and how is it decided who plays who at the beginning of a tournament?

All the entrants are divided up into pools of players with roughly the same number of players in each pool. The TOs (tournament organizers) will try to seed the well-known, established players so that there are just the right number of them in each pool.

There are two types of pools: round robin and bracket. Round robin is when everyone in the pool plays each other. At the end of the round robin, the players with the best records make it to the next round of the tournament, which might be another round of pools or a bracket. Everyone else is eliminated. The number of players who make it out of each pool is predetermined by the TO. Example: Your pool has eight people. Three people will advance from it. You play everyone else in your pool for a total of seven matches. If your record after the round robin is in the top three, you advance into the next round. The placings of the top three (or however many get through) will also matter in seeding for the next round or bracket. Tiebreakers in placings are done using games lost (as in the games lost in sets).

The second type of pool is the bracket pool. You can think of it as a bunch of brackets instead of just one. Again, TOs will try to seed the good players so that there are an even number of them in each pool. Each bracket pool is a double elimination bracket. Essentially, you have two lives. It is a bracket, but you aren't out if you lose once. Everyone starts out in the winner's bracket. If you win a match, you move on in the winner's bracket. If you lose, you get sent into the loser's bracket and fight other people who have lost once in your pool. When you lose in the loser's bracket, you're out of the tournament. Like with the round robin pool, a certain number of people from the bracket pool advance into the next round. For example, the TO may decide to take the final two players in the winner's bracket and the final two players in the loser's bracket to advance into the next round.

What happens when you advance from the bracket pool is determined by the TO. Some tournaments will make it so that if you lose in a bracket pool, you're in loser's bracket for the entire tournament. Others will put you back into winner's bracket in the next round even if you got sent into loser's.

I don't understand winners bracket and losers bracket. Can someone still win even if they aren't in winners bracket? Is there a helpful visual aid someone could link?

I explained winner's and loser's brackets, but there is more. At the end of the tournament, the grand finals will be between the last man standing in the winner's bracket and the last man standing in the loser's bracket. Because the guy in the winner's bracket hasn't lost yet (still has two lives), the guy in the loser's bracket has to beat the winner's bracket guy twice in order to win the tournament. The guy in the winner's bracket only has to beat the loser's bracket guy once to eliminate him and win the tournament. When the loser's bracket guy beats the winner's bracket guy and sends him into loser's, that is known as resetting the bracket.

For visual aids, you should check challonge.com and search for Smash tournaments.

What is the deal with counterpicking? I know about starter stages and counterpick stages but are there any specific rules about them, or is it just loser picks whatever stage he wants? Along with that, how does character counterpicking work?

The process is that the winner bans a certain number of stages that he doesn't want to play on. The loser then picks a stage out of the ones left. The winner then has an opportunity to change his character. Finally, the loser can counterpick a character if he/she so chooses. Then they play.

After Apex I've heard people talking about coaching being a problem in smash 4. Do people really give their friends advice? Shouldn't the game be between the two playing, and not the backseat drivers?

Yes, people do coach. Your opinion that the game should be between two players solely is one that some people agree with. Other people think coaching is fine and is part of Smash as an esport.