Stream Ciphers crypts the plaintext data with a random stream of bit (usually with a XOR because it can be reversed easily). If you have a 128bit data, you will use a 128 bit psedurandom strempseudorandom stream (your key) to encrypt.
Block Ciphers crypts the plain text data a block at a time with the same transformation (based on the key). So you have your 128bit data, the cipher breaks it in blocks (like 4 block of 32 bits) and apply the same transformation to every block obtaining 4 encrypted blocks, that combined will form the final cryptogram.
Of course because of this Block Cyphers are safer but expensive to use in terms of hardware complexity involved. Stream Ciphers are faster and "cheap" but they can be susceptible to security problems if implemented incorrectly.