From man select
int select(int nfds, fd_set *readfds, fd_set *writefds,
fd_set *exceptfds, struct timeval *timeout);
nfds is the highest-numbered file descriptor in any of the three sets, plus 1.
What is the purpose of nfds, when we already have readfds, writefds and exceptfds, from which the file descriptors can be determined?
In "Advanced Programming in the UNIX Environment", W. Richard Stevens says it is a performance optimization:
By specifying the highest descriptor we're interested in, the kernel can avoid going through hundred of unused bits in the three descriptor sets, looking for bits that are turned on.
(1st edition, page 399)
If you are doing any kind of UNIX systems programming, the APUE book is highly recommended.
UPDATE
An fd_set is usually able to track up to 1024 file descriptors.
The most efficient way to track which fds are set to 0 and which are set to 1 would be a bitset, so each fd_set would consist of 1024 bits.
On a 32-bit system, a long int (or "word") is 32 bits, so that means each fd_set is
1024 / 32 = 32 words.
If nfds is something small, such as 8 or 16, which it would be in many applications, it only needs to look inside the 1st word, which should clearly be faster than looking inside all 32.
(See FD_SETSIZE and __NFDBITS from /usr/include/sys/select.h for the values on your platform.)
UPDATE 2
As to why the function signature isn't
int select(fd_set *readfds, int nreadfds,
fd_set *writefds, int nwritefds,
fd_set *exceptfds, int nexceptfds,
struct timeval *timeout);
My guess is it's because the code tries to keep all the arguments in registers, so the CPU can work on them faster, and if it had to track an extra 2 variables, the CPU might not have enough registers.
So in other words, select is exposing an implementation detail so that it can be faster.
I don't know for sure, since I'm not one of the designers of select(), but I'd say it's a performance optimization. The calling function knows how many file descriptors it put in the read, write and except FDs, so why should the kernel figure it out again?
Remember that in the early 80s, when select() got introduced, they didn't have multi-gigaghertz, multi-processors to work with. A 25 MHz VAX was pretty doggone fast. Plus, you wanted select() to work fast if it could: if some I/O was waiting for the process, why make the process wait?