There are many different tests which one can use in order to determine if a sequence of words is a syntactic constituent. The five below are some of the easiest to understand and provide examples for.
1. Topicalization- This is a simple movement of the sequence of words to the front of the sentence.
Example: He went to the mall to buy some shoes. To buy some shoes, he went to the mall.
2. Clefting- This is where the sequence is placed after the words "it is" or "it was."
Example: He bought a pair of shoes with red laces. It was a pair of shoes with red laces that he bought.
3. Pseudoclefting- This is similar to clefting, but is simply used to place emphasis on part of the sentence.
Example: He bought a pair of shoes with red laces. A pair of shoes with red laces is what he bought.
4. Answer fragments- For a sequence of words to be determined syntactic constituent, they must be able to exist as a stand alone fragment (still make sense).
Example: Question: What did you do last night? Answer: Watched TV. (correct) Watched. (Incorrect)
5. Omission- This test examines if a sequence of words can be removed from a sentence without changing the meaning of the sentence.
Example: Sally relaxes during the day on the beach. Sally relaxes during the day. Sally relaxes on the beach. According to the test, both "during the day" and "on the beach" are constituents.
The second link provided below is a PDF which explains other constituency tests.