An IV is essential when the same key might ever be used to encrypt more than one message.
The reason is because, under most encryption modes, two messages encrypted with the same key can be analyzed together. In a simple stream cipher, for instance, XORing two ciphertexts encrypted with the same key results in the XOR of the two messages, from which the plaintext can be easily extracted using traditional cryptanalysis techniques.
A weak IV is part of what made WEP breakable.
An IV basically mixes some unique, non-secret data into the key to prevent the same key ever being used twice.