As other answers have mentioned, NT (the kernel underlying modern versions of Windows) has an equivalent of Unix fork(). That's not the problem.
The problem is that cloning a process's entire state is not generally a sane thing to do. This is as true in the Unix world as it is in Windows, but in the Unix world, fork() is used all the time, and libraries are designed to deal with it. Windows libraries aren't.
For example, the system DLLs kernel32.dll and user32.dll maintain a private connection to the Win32 server process csrss.exe. After a fork, there are two processes on the client end of that connection, which is going to cause problems. The child process should inform csrss.exe of its existence and make a new connection – but there's no interface to do that, because these libraries weren't designed with fork() in mind.
So you have two choices. One is to forbid the use of kernel32 and user32 and other libraries that aren't designed to be forked – including any libraries that link directly or indirectly to kernel32 or user32, which is virtually all of them. This means that you can't interact with the Windows desktop at all, and are stuck in your own separate Unixy world. This is the approach taken by the various Unix subsystems for NT.
The other option is to resort to some sort of horrible hack to try to get unaware libraries to work with fork(). That's what Cygwin does. It creates a new process, lets it initialize (including registering itself with csrss.exe), then copies most of the dynamic state over from the old process and hopes for the best. It amazes me that this ever works. It certainly doesn't work reliably – even if it doesn't randomly fail due to an address space conflict, any library you're using may be silently left in a broken state. The claim of the current accepted answer that Cygwin has a "fully-featured fork()" is... dubious.
Summary: In an Interix-like environment, you can fork by calling fork(). Otherwise, please try to wean yourself from the desire to do it. Even if you're targeting Cygwin, don't use fork() unless you absolutely have to.