The names P-xxx, B-xxx and K-xxx refers to NIST notation.
You can find the names and the curves' parameters in Appendix D of FIPS 186-4.
NIST notation uses P-xxx for curves defined over large characteristics (i.e. prime fields), B-xxx for random binary curves ($a=1,b$ random), while K-xxx is for binary curves of Koblitz form ($a=[0,1],b=1$)
The other names you mentioned, starting with "sec" refers to the SECG SEC1 standard. Here all curves start with "sec" followed by "p" for prime fields or "t" for binary, followed by the size of the field and by "r" or "k" where "r" stands for randomly generated and "k" stands for Koblitz type.
There are correspondence between the two notations. By looking at the documents and the actual curves' parameters, you can find, for example, that:
- P-256 = secp256r1
- P-384 = secp384r1
- P-521 = secp521r1
- B-283 = sect283r1
- K-283 = sect283k1
- B-409 = sect409r1
- K-409 = sect409k1
- B-571 = sect571r1
- K-571 = sect571k1
Note: SEC extends the concept of Koblitz curves also for prime field curves, such as the bitcoin secp256k1. In this context, a Koblitz curve has an additional efficiently computable endomorphism.
The name prime256v1 (the same curves as P-256 and secp256r1) comes from ANSI X9.62.
And as far as I know the brainpool standard is the only one naming the brainpool curves, so their name is unique.