General understanding:
* Comprehend
* Grasp
* Perceive
* Apprehend
* Get
* See
* Realize
* Know
* Figure out
* Make sense of
* Digest
* Absorb
Empathy and emotional understanding:
* Relate to
* Feel
* Sympathize with
* Empathize with
* Get into someone's shoes
Technical understanding:
* Master
* Learn
* Acquire knowledge of
* Be familiar with
* Be proficient in
Informal understanding:
* Get it
* Catch on
* Figure it out
* Wrap your head around it
* See what you mean
The best synonym for "understand" will depend on the specific situation and the level of formality required.