There are two competing notions of the predicate in theories of grammar.
The first concerns traditional grammar, which tends to view a predicate as one of two main parts of a sentence, the other part being the subject; the purpose of the predicate is to complete an idea about the subject.
The second derives from work in predicate calculus (predicate logic, first order logic) and is prominent in modern theories of syntax and grammar. In this approach, the predicate of a sentence corresponds mainly to the main verb and any auxiliaries that accompany the main verb, whereas the arguments of that predicate (e.g. the subject and object noun phrases) are outside the predicate.
English sentences at a minimum must contain at least one word which is a subject and one word which is a verb, unless it's imperative, in which case the subject is understood if it's "you". The verb and everything attached to it is the predicate.
So if it doesn't have a predicate, and it's not imperative, it's not a sentence. If you have a standalone group of words with a missing subject or predicate, that would be a sentence fragment.
Now, sentences can consist of one or multiple subject-predicate pairs - if they have more than one, typically (but not always) they are separated by "linker" words such as that, but, etc. These are called clauses.
You also can have a group of related words without a subject or predicate that's part of a bigger sentence--that would be a phrase. Prepositional phrases are common - to the window, to the wall, etc.
Google's definition of phrase (and clause) is pretty good:
A phrase is a group of related words that does not include a subject and verb. (If the group of related words does contain a subject and verb, it is considered a clause.) There are several different kinds of phrases.