A parse tree represents the structural construction of a sentence with respect to the grammar of the language in question.
For example, we could construct a toy grammar for the English language using the subject-predicate-object structure, as
Sentence ::= NP VP | NP VP NP NP ::= Noun | DT Noun VP ::= Verb
using the conventional names of NP for Noun Phrases, VP for Verb Phrases and DT for Determiners like the or an.
Using this grammar, we can describe sentences like John sleeps or the dog eats the cake. For John sleeps, we can use the first rule, which states that a Sentence can be an NP followed by a VP. Using the third rule, an NP can be just a noun, such as John, and similarly, according to the fifth rule, a VP can consist of simply a verb, such as sleeps.
Because Sentence is split into NP and VP, which are then further specialized into Noun and Verb respectively, it makes sense to draw this derivation as a tree where each grammatical entity is connected to the entity it is derived from. The parse tree for the example sentence John sleeps would thus be:
Sentence / \ NP VP | | Noun Verb