To understand language, a child must understand the meaning of word parts (a.k.a. morphology) and individual words within the language (a.k.a. vocabulary), but more than that, a child must understand that words are arranged in phrases, sentences, and discourse in meaningful ways. The child must understand how to use language to communicate complete and meaningful ideas.
Semantics is a general term that just refers to "meaning." Vocabulary specifically refers to the meaning of isolated words, and morphology specifically refers to the meaning of word parts, but semantics can generally be applied to the meaning of word parts, whole words, sentences and discourse.
There are several ways to assess semantics at each of these levels, but one common thread involves the question of whether the items on the test are presented in written form. If the child is expected to read the items, the test becomes more of a decoding test than a test of semantics.
Although the items should not involve printed text, it is very common to use pictures in semantics assessments (teachers should, of course, consider the cultural relevance and sensitivity of the pictures they use). A child might be asked to provide a name for pictures as a test of expressive vocabulary, or to match spoken words with pictures as a test of receptive vocabulary. A test of semantics at the larger-than-word level may involve asking a child to arrange a series of pictures to reflect a logical sequence of events.
Another common assessment involves asking a child to provide a word that best matches a definition presented (orally) by the teacher as a test of expressive vocabulary, or to ask a child to provide a definition of a word as a test of receptive vocabulary. Similarly, a test of vocabulary knowledge could require that the child be familiar with several words in order to answer each item correctly. For example, the child could be asked to select a word which does not belong in a group of words (e.g. THREAD, STRING, ROPE, KNOT). In this sort of assessment, the child must know the meaning of most if not all of the words in each item in order to be successful.
Similarly, a child might be asked to provide a synonym or an antonym for words, which is a test of both receptive and expressive vocabulary. Again, in this case, more than one vocabulary word is being tested at a time - the child must know the meaning of the test item, and must know another word which either has the same meaning or an opposite meaning.
Morphology assessments often involve asking a child to describe how a word's meaning changes as parts of the words are changed. For example, a child could be asked to break compound words into their component parts and to describe the meaning of those component parts (e.g. DAY-BREAK, BASE-BALL, HEAD-ACHE). Or, a child could be asked to describe what happens when affixes are added to words (as in SKIP versus SKIPPED) and to explain those affixes (What do UNWRAP and UNTIE have in common?). Similarly, a child's appreciation of morphology can be assessed by asking the child to describe how words with similar parts are related (e.g. EARACHE, EARRING, EARDRUM).
Semantics assessments at the larger-than-word level usually depend on identifying words or sentences that do not make sense in the context (e.g. "Billy had a dog. He loved his dog. His fish was orange. His dog could fetch a ball."), or they depend on the child identifying logical inconsistencies (e.g. "Billy's dog could fetch and he could roll over. He was a good dog, but he didn't know any tricks.").