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PPPoE stands for point to point protocol over ethernet. It is a method of encapsulating your data for transmission to a far point.
PPPoA is PPP over ATM.. from a software point of view, this is very similar to PPPoE .. in this case though, a DSL modem pumping ATM is internal to the computer, rather than being a short ethernet cable away.
Originally designed for dialup lines, it is being used by DSL providers to solve the problems they get managing an open DSL network, viz: IP address shortages, broadcasts not meant for you appearing on your local IP address (because you are on a giant ISP centered virtual net), and other (mainly ISP-end) difficulties inherent in large bridged networks.
Some heavy duty DSL back-end suppliers are promoting PPPoE as a way of helping ISPs cope with these difficulties. Redback, for example, is promoting PPPoE (PPP over ethernet), with a lot of success.
With a PPPoE setup, your computer (or in some cases, DSL hardware) needs to run a PPPoE protocol stack (software). This will be recommended by your DSL supplier. One of the more common PPPoE stacks is WinPOET. There may be difficulties combining NAT and PPP or PPPoE implementations.. it generally makes things more complex. Check the FAQ for common problems with PPPoE, DHCP and other issues. If you have a choice, try to get a bridged static IP.
Your ISP usualy dictates witch you use, so most people have no choice. The only way that you'd get the choice, is if you have different ISP's offering different choice.