The -te form has many uses; here there are just some of them:
1) It's a way to connect two sentences (a kind of equivalent expression for "and").
2) Equivalent to English "-ing".
3) -te+iru: to describe an action that is now happening or that is not already finished. [There are similar grammatical structures with verbs like "shimau", "ageru", "yaru", "miru"... but I guess they should have their own section]
4) Replacing an adverb.
5) Motte+iku/kuru: to bring, to take, to carry something.
6) Present perfect in some cases.
When the -te form is applied to adjectives finished in -i, it has also a copulative function: joining two or more adjectives by adding -kute instead -i.
天気が悪くて、たいへん困りました。 The weather was bad, and that bothered us.
あの映画は短く面白いです。= あの映画は短くて面白いです。 That film is short and amusing.
[I must apologize for the possible mistakes this note may have. English is not my native language]
It's a JLPT level 4 grammar point (both the -て and -くて forms, btw), and not a level 3 one; Only some of the -て + verb forms are part of the JLPT level 3 grammar requirements
Note, many, many, other uses of [vte].
Hmm, not the most natural example sentence in the world ^^v
Also -i adjectives join with -kute -くて The copula has the 'te' form of で.
Verbs with auxillary verbs and auxillary adjectives use the -te form. e.g. やってみてください。 やる(in -te form, to do) みる (in -te form, to try) ください (request). All together Please try to do it.