...a women named Mariam who claim to be the widows of either the late Nigerian strongman Sani Abacha or the deceased Zairian dictator Mobutu Sese Seko. They need help tapping into some Swiss bank accounts.
Essentially Mariam or one of her compatriots will tell you that they are sitting on a pile of money which they need to move across borders with the aid of a "foreign based intermediary" (you), and once you pay them a certain amount of your own money upfront in a show of good faith, you will be awarded to the tune of millions of dollars. However, if you ever actually appear in Nigeria, cash in hand, you are likely to be kidnapped and held for ransom, or murdered if you run low on funds.
The infamous Nigerian letter scam has an old and storied history, which the linked article from Slate discusses admirably. I remember 15 years ago when these things used to come over the fax line, about maybe one or two per month. Even then if you reported them to your local police they would basically ignore you (after telling you NOT to respond). But since the rise of the Internet, costs of running the scam have plummeted and, whereas they used to be the provenance of a few well organized gangs, if you walk into a Nigerian Internet Cafe these days most of the people you are likely to meet are running 419s (after the relevant section of the Nigerian penal code). In fact, 419 scams are being credited with driving the "wiring" of Nigeria in the same way that the porn industry drove most of the early Internet advances in North America. Nigerian letter emails count for the second most common form of spam on the Net.
There's a good website devoted to Nigerian E-mail scams here and here. Often, if you make contact with a 419 scammer, they will send you a horrible fake photo to convince you that they are in fact the "important government official" they claim to be (see above). Here is a page of some of the funnier ones.
Finally, if you are interesting in attempting your own version of the scam (while most originate in Nigeria, a few have apparently been traced to Canada), they tend to work like this: