A deterministic wallet is a wallet where private and public keys are all derived from a starting seed value. This could be a long passcode/password, or be a random series of letters and numbers.
A typical wallet creates private and public keys on demand for the user. This means that the wallet needs to be backed up frequently, otherwise coins may be lost. Also, having multiple machines with wallets on them means it is difficult to manage all of your coins together.
A deterministic wallet can be backed up by simply copying the starting seed value to a secure location, and this only needs to be done once. If the wallet ever gets lost, all private and public keys can be regenerated from the initial seed.
Also, multiple devices could host the same wallet based off of the same seed and automatically stay in sync with eachother. Non-critical information such as address books would need to be stored and copied between wallets.
If the initial seed value was either guessed or taken, the attacker could take all of the coins from the wallet. Also, they could retain that seed value, and wait until some future date to take all of the coins.
Passwords vs Random Strings
The passcode/password has the benefit of being memorizable by the user, but at the expense of being either forgotten, or weak enough that the password could be guessed or brute forced. If a user used a password such as abc123, and an attacker might simply go through a list of common passwords, create wallets for them, and see if the public addresses match anything currently in the blockchain.
A long string of letters and numbers would be a way to prevent a brute force attack. This has the drawback of having to be actually stored somewhere. If this code was ever lost, the wallet would be lost forever.
Types of deterministic wallet in use
Each implementer of deterministic wallets should make sure that this article leads to a publicly available reference describing how to reconstitute the deterministic wallet from its seed.
Type 1 deterministic wallet
A Type 1 deterministic wallet is created from a string. Simply take SHA256(string + n), where n is an ASCII-coded number that starts from 1 and increments as additional keys are needed. This simple type of wallet can be created by Casascius Bitcoin Address Utility.
Type 2 deterministic wallet
Not sure on the details, but mention was made of a "type-2 deterministic wallet" in BIP 0032 and credited to Gregory Maxwell, so this is a placeholder to describe that implementation. The relevant form topic is here.
BIP 0032 deterministic wallet
Described in BIP 0032 (currently a draft) and described as a hierarchical deterministic (HD) wallet, a BIP 0032 deterministic wallet allows sharing smaller deterministic wallets that are subportions of a larger one.
Electrum deterministic wallet
Electrum implements a Type-2 deterministic wallet format based on a 128-bit seed. It uses a word list and converts the seed to a series of words as an aid to help the user record the seed.
Armory deterministic wallet
Armory has its own Type-2 deterministic wallet format based on a "root key" and a "chain code." The Armory client has a "Paper Backup" screen that allows the user to print these data or copy it down by hand. Earlier versions of Armory required backing up both the "root key" and "chaincode," while newer versions start deriving the chaincode from the private key in a non-reversible way. These newer Armory wallets (0.89+) only require the single, 256-bit root key.