The card has data that is arranged in sectors. To access sector data you need to provide a key (that's what happens with Login). Depending upon the key, you may be able to just read the data or read and write data back to the card. Key distribution depends upon your application (for example, access control is totally different than having a transport wallet like the Oyster Card).

The mifare cards use what at first inspection appears to be a complex access control system but when you get familiarized it's not that hard. However, in order to get more info you need the ACR SDK documents. I don't know what you've purchased but I suspect that if you only got an ACR contactless reader/writer you didn't get access to the documents. If that is so, you need to purchase an ACR SDK which comes complete with a contactless reader/writer and the SDK CD. That includes the "Mifare Standard Card IC Functional Specification" (without which you have no chance of figuring out the Mifare card scheme) and the "ACR Contactless Reader/Writer API". There are also some other documents but these are the most important ones.