What are i-vectors and x-vectors in the context of Speech Recognition?
The i-vectors and x-vectors share the ability to represent speech utterance in a compact way (as a vector of fixed size, regardless of length of the utterance).

The extraction algorithms of i-vectors and x-vector are quite different. The x-vector concept is newer and the name of the method is similar to "i-vector" to suggests that this representation can be used instead of i-vectors in state-of-the-art speaker (or language) recognition systems.

There are many good sources explaining how i-vectors work, I recommend this presentation from Najim Dehak (who is the inventor of the i-vector concept).

The x-vectors are embeddings extracted with DNN, but more detailed explanation can be found here (there is also a youtube lecture explaining the concepts presented in this paper).