Tú come o tú comes

Tú come o tú comes
 

Ive learned that when you talk to a person, you usually concatenate 's' to the verb. But I got to see sentences which contain an addressing to a person, and the verb was without 's'. For instance: 'puede traer la cuchara' or 'disfruta el fin de semana'
What are the rules for that? Is that some kind of slang? Or a polite way to talk with someone?

There are two forms of address - formal and familiar. The first is conjugated using a third-person ending, which has no 's'. The second is conjugated using a second-person ending, which does have an 's'. In your first sentence, you have a case of a third-person ending.

Study up on second- and third-person endings (using tú and usted, respectively). The endings will change based on tense/mood. You can go here or here to see conjugation charts for disfrutar. Take note of ALL the possible endings! There are three persons, both singular and plural.

Your second sentence is an example of the imperative mood (modo imperativo) being used in the second person.