How is it possible to connect a mainframe with a PC running Windows?.Is it still regarded a mainframe as a computer with a high processing power running Unix or IBM System360
We do it every day. You can connect to the OMVS/USS (UNIX system services) subsystem using standard telnet or ssh (if the mainframe has had that set up) but, to access the z/OS interface (TSO/E or ISPF), you'll need a 3270 emulator such as x3270 (we use IBM's Personal Communicator product because it's incredibly powerful).
The mainframe is in no danger of disappearing anytime soon. There is still a large amount of work being done at the various IBM labs around the world and the 64-bit z/OS (nee MVS, nee OS/390, ...) has come a long way.
It's still used in the big corporations as it's the only thing that can be trusted with their transactions - the z in System z means zero downtime and that's not just marketing hype. The power of the mainframe lies not in it's CPU grunt (the individual processors aren't that powerful but they come in books of 54 CPUs with hot backups, and you can run many books in a single System z box) but in the fact that all the CPU does is process instructions.
Everything else is offloaded to specialist processors, zIIPs for DB2, zAAPs for Java workloads, other devices for I/O (and I/O is where the mainframe kills every other system, using fibre optics and very large disk arrays). I wouldn't use it for protein folding but it's ideal for where it's targeted, massively insane levels of transaction processing.
z/OS has a UNIX subsystem and z/VM can run multiple copies of z/OS, other z/VMs and even zLinux (RHEL & SLES) - I've seen a single z800 box running thousands of instances of RHEL concurrently. This puts all the PC manufacturers 'green' claims to shame and communications between the instances is blindingly fast with HyperSockets (TCP/IP but using shared memory rather than across slow network cables (yes, even Gigabit Ethernet crawls compared to HyperSockets (and sorry for the nested parentheses :-))).
It runs Websphere Application Server and Java quite well in the Unix space while still allowing all the heritage stuff to run as well. In fact, mainframe shops need not buy PC-based servers at all, they just plonk down a few zLinux VMs and run everything on the one box.
And recently, there's talk about that IBM may be providing xSeries (i.e., PCs) plugin devices for their mainframes as well. While most mainframers would consider that a wart on the side of their beautiful box, it does open up a lot of possibilities for third-party vendors. I'm not sure they'll ever be able to run thousands of Windows instances but that's the sort of thing they seem to be aiming for (one ring to rule them all? :-).
If you're interested, there's a System z emulator called Hercules which I've seen running at 23 MIPS on a Windows box (barely enough grunt to start ISPF for your mainframers out there) but it runs the last legally-usable MVS 3.8j fast enough to get a feel.
To provide a shameless plug for a book one of my friends at work has just written, check out What On Earth is a Mainframe? by David Stephens (ISBN-13 = 978-1409225355). I found this invaluable since I came from a PC/UNIX background, and it is quite a paradigm shift.