The 3 main codecs that most listeners are familiar with are SBC, AAC and aptX:

SBC -  Subband Coding -  The mandatory and default codec for all stereo Bluetooth headphones with the Advanced Audio Distribution Profile (A2DP). It is capable of bit rates up to 328 kbps with a sampling rate of 44.1Khz. It provides fairly good audio quality without requiring a lot of processing power to encode or decode. However, the audio quality can be a bit inconsistent at times. This is especially noticeable with a cheap Bluetooth transmitter.

AAC - Advanced Audio Coding - Similar to SBC but with a better sound quality. This codec is mostly popular with Apple's iTunes platform and some other non-wireless applications. However, it's not very common, especially for headphones.

aptX -  A proprietary and optional codec designed by CSR. It's ideal for demanding audio applications since it encodes audio more efficiently and at a slightly higher rate than SBC. There are also two additional variation aptX(LL) and aptX HD that either drastically reduces the latency of the connection or significantly improves its audio quality. However, it's a bit limiting as both the Bluetooth transmitter and receiver must have aptX or its variations for the codec to work.

Latency

Codecs have a bigger impact on latency (learn more about this test) than on sound quality for most listeners. The default SBC connection typically has more than 100 ms of latency which is noticeable when watching videos and may be severe enough to ruin your gaming experience. 

To fix some of the sync issues caused by latency, CSR developed the aptX and subsequently the aptX-Low Latency codec. Regular aptX does somewhat improve latency due to its more efficient encoding algorithm than SBC. However, aptX-LL has the most noticeable impact on latency.

Conclusion

Codecs are the algorithms that compress data for easier and faster transmission. Better encoding and decoding algorithms mean less lossy transmission which can help with audio quality. We've noticed that codecs have a bigger impact on latency than on audio quality.