Sentence:  The terminology "<WSD>splitting field</WSD> of a K-algebra" is motivated by the same terminology regarding a polynomial. A <WSD>splitting field</WSD> of a K-algebra A is a field extension K\mapsto L such that A\otimes_K L is split; in the special case A=K[x]/f(x) this is the same as a splitting field of the polynomial f(x). 
Meaning of the word marked with <WSD> tags:  Given a finite-dimensional "K"-algebra (algebra over a field), an extension field whose every simple (indecomposable) module is absolutely simple (remains simple after the scalar field has been extended to said extension field).