。
I could hear our dog barking all night.
Literal
All-night our dog [subj-が] is-barking-[nom-のが] was-audible.
~のが聞こえる ('X could be heard') uses the nominalizer の to embed a clause as the object of a perception verb. 聞こえる is the intransitive 'be audible, can be heard' — distinct from the transitive 聞く ('listen to') and the potential 聞ける. うち here means 'our (household's)' — the word 家 (house) extended to mean 'our place, our family.' 吠える ('bark') is the standard verb for canine vocalization.