The main reason for the switch from 32-bit is to get past the limit on the amount of memory a 32-bit processor can access.
32-bit vs 64-bit: Memory
A 32-bit processor can access a maximum of 4GB of memory, which isn't enough these days. It might be fine for basic tasks, but if you want to run multiple programs, have 35 tabs open in various web browsers or open 30 high-resolution photos for editing at once, 4GB isn't going to cut it. (And don't forget that 32-bit Windows can access only around 3.2GB even if 4GB is installed.
PCs and laptops and their operating systems are all 64-bit these days. 64-bit versions of Windows 8 can access 8TB of RAM http://msdn.microsoft.com/en-gb/library/windows/desktop/aa366778(v=vs.85).aspx, while server versions can access 128TB.
However, you'll struggle to afford such enormous amounts of memory, let alone install it into today's motherboards which typically top out at 32GB over four slots.
x86 vs x64: What does the jargon mean?
This somewhat misleading term is the equivalent of 32-bit vs 64-bit. x86 (or x86 architecture) has long been a common way to refer to the family of processors ending in 86, Intel's most successful line of processors. The first was the 8086, and many people will remember the 286, 386 and 486 processors in early PCs from the mid-1980s to 90s.
All are 32-bit, which is why x86 has become interchangeable with 32-bit.
Likewise, x64 has become interchangeable with 64-bit (and is even used by Microsoft), but doesn't really make sense as there aren't lots of processors with numbers ending in 64. Instead, it is merely shorthand for 64-bit.
64-bit vs 32-bit software and compatibility
Memory isn't the only reason to have a 64-bit computer. 64-bit processors can deal with data in 64-bit chunks, which is obviously more efficient than moving it around in 32-bit words.
If your PC is running 64-bit Windows, you'll probably find a Program Files (x86) folder on your hard drive. This stores 32-bit applications, while the other 'Program Files' folder contains all the 64-bit apps you've installed.
64-bit systems can run 32-bit programs, as they're backwards compatible. It doesn't work the other way around, though: a 32-bit computer cannot run 64-bit Windows or 64-bit programs.