Essentially, verbal agreement is a construct whereby a verb used with a particular pronoun must take one form, and another form when used with another pronoun. French exhibits verbal agreement: in the phrase je suis I am, the verb être to be has agreement with the pronoun je I, and does not appear in the form être, but in the form suis. More transparently, manger to eat exhibits agreement with nous we/us in nous mangeons we eat.
Verbal agreement allows a language to do away with pronouns: Spanish, for instance, no longer has need for free pronouns in sentences, even though French, its close relative, still does: the Spanish equivalent to the French je suis is soy, from an original form yo soy.
Verbal agreement also frees up word-order. In Basque, word order is nominally Subject-Object-Verb. However, its extensive agreement (coupled with its complex case system) allows the practical word order to be almost completely free, with all permutations of Subject-Verb-Object being permitted.
Verbal agreement is not necessarily just for the subject of the verb, although this is the case in the majority of Western European languages. Ubykh exhibits verbal agreement for the subject, direct object, indirect object, benefactive and ablative objects (a.w3.s.xe.n.t'u.n you gave it to him for me), and Basque can show agreement not only for subject, direct object and indirect object, but it also on occasion exhibits agreement for listener: autoa digute means he brought the car (neutral agreement for listener), but autoa zigunate means he brought the car (agreement for female singular listener).