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BSD licences
First slightly annoying thing: there isn't just one BSD licence.
The two you're most likely to meet are the BSD 2-Clause Licence and BSD 3-Clause Licence. Both are permissive licences, and both are fairly short.
What do they mean in normal English?
Very roughly: you can use, change and redistribute the code, including in commercial and closed-source software, but you need to keep the required copyright, licence and disclaimer notices.
BSD 2-Clause
The 2-Clause licence allows redistribution in source and binary forms, with or without modification, as long as its conditions are met.
If you redistribute the source, the copyright notice, conditions and disclaimer need to stay with it. If you redistribute a binary, those notices need to appear in the documentation or other materials supplied with it.
BSD 3-Clause
The 3-Clause version adds another condition: you cannot use the copyright holder's name, or the names of contributors, to endorse or promote products derived from the software without specific prior written permission.
So if Nathan Tech released a library under BSD 3-Clause, somebody could potentially use the library in their product. They could not then decide that "Nathan Tech approved this product" sounds nice on the advertising.
Can BSD code go into closed-source software?
Yes. These are permissive licences.
They do not generally say that the rest of a program has to become open source merely because BSD-licensed code is used inside it.
Can it be used commercially?
Yes. Commercial use and redistribution are allowed under the licence conditions.
Do I lose my copyright?
No. Licensing your code is not the same as giving your copyright away.
You are granting other people permissions under particular terms.
What about warranty?
The BSD licences include an "as is" disclaimer and disclaim listed warranties and liability.
BSD or MIT?
For a lot of small projects, BSD 2-Clause and MIT have a very similar overall aim: broad reuse with notice requirements.
BSD 3-Clause adds the non-endorsement condition. If you're choosing between these licences, read the actual text of each. They're short enough that this is one of the nicer licence comparisons!
An example
You release an audio library under BSD 3-Clause. Another developer includes it in a commercial game.
That can be allowed. They need to follow the notice requirements, and the licence does not give them permission to use your name to imply that you endorse their game.