Common and technical use
There is a difference between the common use of the term phrase and its technical use in linguistics. In common usage, a phrase is usually a group of words with some special idiomatic meaning or other significance, such as "all rights reserved", "economical with the truth", "kick the bucket", and the like. It may be a euphemism, a saying or proverb, a fixed expression, a figure of speech, etc.
In grammatical analysis, particularly in theories of syntax, a phrase is any group of words, or sometimes a single word, which plays a particular role within the grammatical structure of a sentence. It does not have to have any special meaning or significance, or even exist anywhere outside of the sentence being analyzed, but it must function there as a complete grammatical unit. For example, in the sentence Yesterday I saw an orange bird with a white neck, the words an orange bird with a white neck form what is called a noun phrase, or a determiner phrase in some theories, which functions as the object of the sentence.
Theorists of syntax differ in exactly what they regard as a phrase; however, it is usually required to be a constituent of a sentence, in that it must include all the dependents of the units that it contains. This means that some expressions that may be called phrases in everyday language are not phrases in the technical sense. For example, in the sentence I can't put up with Alex, the words put up with (meaning 'tolerate') may be referred to in common language as a phrase (English expressions like this are frequently called phrasal verbs) but technically they do not form a complete phrase, since they do not include Alex, which is the complement of the preposition with.