To be fired, means you did something wrong. You may be fired without cause, which means they don't want to document why you were let go, or fired with cause, meaning they document fully why you were let go.
Without cause is generally political move to give you a place to go. It's a bit of a risk for a company to fire without cause, so it's pretty rare, but it does happen when the company wants you gone, and would rather do it faster then documentation requires. For example, you really piss your boss off, and while they could write you up three times with 30 days in between for minor infractions then fire you, they just want you gone. You as an employee may accept this, because you found a new awesome job and want to go there, and don't want to have to wait 90 days to get fired, or want to breach your contract.
Another good example of this (fired without cause) would be when you have good terms with your employer, but you find a much better job offer they can't match. However your contract states that your there for a certain amount of time or you loose a bonus or some such. Your current employer may fire you without out cause so that you can take the new job, effectively releasing you from your contractual obligations.
Fired with cause is easy. You did something wrong. Maybe performance, maybe behavior, maybe both. But you, individually did something wrong.
By far fired with cause is more common and fired without cause is risky for the company. If you get fired, you can almost be certain it's with cause.
Laid Off means that your job (not you) was reduced. It's not uncommon, and there are a lot of reasons to do it. Usually it's motivated by money. For example if you have a worker making $20k and a worker doing the same job for $40k You might lay off get $40k worker and hire 2 new $20k workers.
Another example is a company that needs to reduce their expenses may layoff half their employees to do so.
This happens a lot in support and retail jobs around the holidays. There is a big hiring phase in October or November and in January or February a big "Lay off".
What it means for benefits
Different states have different rules but in general lay off, or fired without cause will get more benefits for longer. Fired with cause will get little to no benefits. This will depend a lot on the local laws.
What it means for job seeking
Fired with cause you will 100% have to explain. You will need to identify what you did wrong and why you won't do it again. It can be very difficult to recover from a fired with cause. Generally the company you were fired from will give you a negative reference. Keep in mind that most US companies will only confirm employment, attendance, duties, and termination reason. So you new job prospect calls your old job and the old job may say "Employed from 2010 to 2017, good attendance, server admin, fired for sexual misconduct" and that's it. They likely won't explain more then that.
Fired with out cause you will need to explain, but as it's generally mutual you may still get a positive reference. "Employed from 2010 to 2017, good attendance, server admin, let go under mutual agreement" This may actually be a good thing, or bad, depends. It's certainly a flag for the interviewer, but one that can be explained away.
Laid off, generally needs no explanation. You past employer probably won't give one, and you just generally say something like, "I was laid off due to budget cuts." It happens and is a normal part of business. It depends a lot on your job though. If your an "extra" employee that is easy to out source, it probably happens all the time, while "core" employee that is very difficult to outsource, it may be very rare. When I worked as a developer for a call center years ago, almost every agent (phone call taker) had been laid off several times in their carriers. They would get laid off at one location and just move to another. Or laid off at one company and move to another. For example a T-Mobile contract would leave company A and go to company B. 90% of the staff on that contract would be laid off, and just go get a job at company B, then when the T-Mobile contract eventually moved back to company A, they would be laid off at company B and re-hired at company A.
When people talk, they generally use Laid off to mean fired or laid off. It's less embarrassing that way. But you need to make sure you have it 100% correct when your going through interviews. Saying you got laid off when you were really fired is a automatic "ban" from the interview process.