Dial-up modems might be slow but they are not going away any time soon. It’s the only kind of Internet connection you can get on the road. Additionally, it is the only service financially available to many.

One of the big problems with dial-up solutions is not the speed; it is the fact that more and more applications are being written for 'always on' internet connections. Programs like to 'call ' to see if updates are available - not to mention all the spyware out there trying to report back to it's manufacturer. Many people find their modem dialing out at random times because some stray piece of software is running somewhere and trying to connect to some internet resource. 

Many broadband routers have serial ports that allow the connection of an external serial modem. The analog modem is considered a 'backup' to a broadband connection but can be operated without the broadband connection. The big problem with this solution is the lack of control over the hang-up process. You must go to the routers admin page to hang the phone up or wait for a period of inactivity. If you have a dedicated phone line for your modem communications then this might be ok.  If not, a using a broadband router to share a modem is not a good idea.

Note, most modems these days are WinModems - even the external modems. This means the modem uses your computer to do most of the processing. Routers will NOT work with WinModems.

Software :

For Windows: Built into most Microsoft operating systems (Win98SE+) is a connection sharing utility called Microsoft ICS (Internet Connection Sharing). Like broadband routers, it relies on NAT (network address translation) to allow all your computers to connect to the internet using a single internet connection. Although not robust, it is free.  Microsoft ICS

Other Solutions  

There are a variety of software based internet sharing solutions and almost all of them work with modems.  Software Internet Connection Sharing