To what extent do domestic animals understand language?
Yes, they can understand some words and even simple sentences.
Here is what Animal Planet says:

Studies show that the average dog can understand about 165 different words, in some cases more if you make a point of training them. This includes the basic commands such as &quot;sit,&quot; &quot;stay,&quot; and &quot;go,&quot; as well as a range of other terms, assuming they're tangible words and not abstract ideas.

Here are some supporting studies:

Some domestic dogs learn to comprehend human words, although the nature and basis of this learning is unknown [1].
A female mongrel dog, Sofia, was initially trained to respond to action (point and fetch) and object (ball, key, stick, bottle and bear) terms which were then presented as simultaneous, combinatorial requests (e.g. ball fetch, stick point). Sofia successfully responded to object-action requests presented as single sentences, and was able to flexibly generalize her performance across different contexts [2].


References:

Tempelmann S, Kaminski J, Tomasello M (2014) Do Domestic Dogs Learn Words Based on Humans’ Referential Behaviour? PLoS ONE 9(3): e91014. doi:10.1371/journal.pone.0091014
Ramos D, Ades C (2012) Two-Item Sentence Comprehension by a Dog (Canis familiaris). PLoS ONE 7(2): e29689. doi:10.1371/journal.pone.0029689