What class of languages is recognized by finite-state automata with $k$ heads?
This model is one of the standard models in automata theory and it has been examined by some researchers. 

The references given in the first comment are very good starting points. 

When the head is two-way, the classes of languages recognized by such models are identical to logarithmic-space classes. However, when the head is one-way, then, up to my knowledge, we do not have a similar exact characterization, but, we have certain incomparable results and some hierarchies based on the number of heads. 

If you are interested in, I recommend you to check alternating, probabilistic, and quantum versions of multi-head automata, as well. Such models can be quite interesting even when using a single head, as the computation is split in into different paths, and then, in each path, the head may access different part of the inputs.

Some generic references:


Turing Machines with Sublogarithmic Space - https://books.google.lv/books?id=vNPuNOo9BUYC


Alternation


Viliam Geffert - https://dblp.uni-trier.de/pers/hd/g/Geffert:Viliam


Probabilistic computation


Ioan I. Macarie - https://dblp.org/pers/hd/m/Macarie:Ioan_I=


Probabilistic and quantum computation


Rusins Freivalds - https://dblp.uni-trier.de/pers/hd/f/Freivalds:Rusins
Abuzer Yakaryilmaz - https://dblp.uni-trier.de/pers/hd/y/Yakaryilmaz:Abuzer


Related models: multi-counter automata and automata using pebble.