A CODEC is an analogue-to-digital converter and a digital-to-analogue converter combined in a single chip.
This means that one single chip can handle playback and recording of sound - it makes the design of sound-cards simpler, because you only have to use one chip, make one set of digital connections, and you don't have to design special logic or programs to cope with the ADC and DAC running simultaneously.
Some CODECs have built in amplifiers (some have microphone pre-amplifiers only, some have headphone amplifiers as well). Some don't.
An audio amplifier boosts an analogue signal - it can be used for boosting very weak signals (e.g. microphone signals) so that they can be accurately processed by an ADC, or they can be used for boosting a signal so that it is strong enough to power speakers or headphones (if you just connected headphones to a DAC, the DAC wouldn't have the strength to produce much sound).
Very crudely, the budget ones are designed for budget products, so tend to have integrated amplifiers - because it saves having to use a seperate amplifer chip and circuit. The high end CODECs tend not to have amplifiers, as it is expected that sound-systems using expensive high-end CODECs will also want to use high-end amplifiers.