About 56k modems
There are many makes and models of modem, and these can make a big difference to the way you connect. When you dial up your ISP, you connect to a bank of modems (known as a modem rack). There are many manufacturers of these, and the make of the ISP's racks can affect your connection.

First, the speed. All modems up to 33600 bps use similar technology, so should connect happily to any other modem. But this is not always the case - some modems don't connect well to some racks. Many of the better ISPs use different makes on different dial-up numbers, so it is worth experimenting if your ISP has more than one number. Ask them for information.

Modems which are capable of 56000 bps fall into two distinct (and incompatible) types. USRobotics (USR) invented X2 technology, while K56flex is the result of a collaboration between Rockwell and Lucent Technology. Both have advantages and disadvantages, both have been largely superseded by the new international standard, V90. Recent models of both types are software-upgradeable (flash upgrade) to the V90 standard. Both use a broadly similar principle.

In an "ordinary" modem, an analogue signal is converted to a digital signal for the phone system. The ISP's modem converts it back to analogue, and here is the 33.6kbps speed limit.
In a K56 modem, the ISP's racks accept the digital signal without any conversion, hence the speed increase. For the same reason, you need your ISP to support whichever 56k system you are using. Also, you need a fully digital phone network. If your signal passes through an analogue exchange, once again the speed advantage is lost.
A simple test for a digital exchange (in the UK) is to dial 1471. If this returns the number of the last person to phone you, you have a digital line.
It is possible to get 56k connections on an analogue exchange, provided there are no other analogue switches between it and the ISP. But in my experience these connections are unreliable.