An address is the Bitcoin equivalent of a bank account. A Bitcoin address starts with a 1 or a 3, and is 27-34 alphanumeric characters in length. The address is generated by the public key.
When sending Bitcoin, you specify a recipient address. Bitcoin cannot be sent without the private key for the sender's address.
Standard Bitcoin addresses begin with a "1", and multi-signature addresses with stronger security begin with a "3". With bitFlyer, you can acquire a multi-signature address for free.
Bitcoin addresses are in a format called "Base 58." Base 64 is well-known in the world of the web. Base 58 is similar, but uses 58 binary data characters.
It is different from Base 64 in that it excludes digits that are easily confused, such as l (lower-case L) and 1 (the number one). For example, lower-case "L" cannot be used.
It is impossible to ascertain someone's public key (private key) from their Bitcoin address, so it is not possible to create a Bitcoin address without a private key. For example, the following is a valid address: 1BitCoinDescriptionAddressadTvGDH
It is possible to actually make payments to this address (although no one has the private key to this address, so there would be no way to get the Bitcoin back.)