Smartphones are now in fact mini computers so they have all the same components as a "normal" computer.
So they have a CPU, memory, "disk" space, a screen, and an operating system (like we have Windows on many of our computers and laptops).
Smartphones fall into three main "camps"
1) Apple iPhones that run iOS software. Apple phones and iOS software are both made by Apple so tightly integrated, but can be expensive. Nobody else runs iOS.
2) Android phones. Android software is made by Google and they make it available to lots of companies so you can buy a Samsung Android phone, a Sony Android phone, a HTC Android phone and so on.
3) Windows phones. Windows is of course made by Microsoft and they are now trying to make Windows look/work the same across all devices in Windows 8. Runs mainly on Nokia phones (as they now own them) but also run on other companies phones.
There is also Blackberry but they are in such a mess I would leave them well alone.
When it comes to phone specs I would not get too hung up on the details, as a simple rule the more expensive the phone the better the hardware components (not 100% true but a reasonable basis to go on).
Important thing is physical size of phone and screen size.
The cheaper phones tend to have smaller screens with less resolution (and less memory), so if you plan to use it on the internet you wont see much on a small screen.
The new Samsung Galaxy S4 for example has a screen resolution of 1920x1080 pixels which is the same as the 24" screen I use on my desktop. Cheaper phones will of course have a smaller resolution.
If you want the "best" phone then Which magazine said the Samsung Galaxy S4 was the best smartphone they have ever tested, though it wont come cheap. The Apple iPhones have always done well on Which tests and have always been "top" until the S4 came out.
With so many phones and so many different contracts then buying a smartphone is a bit of a nightmare.