て form in Japanese is a form of verb that allows you to do various things.
The first thing you need to know is how to turn the dictionary form of a verb into the て form. How to do this depends on how the verb ends.
There are two types of verb. RU verbs, and U verbs. RU verbs usually end in る while U verbs usually end in other things, though there are exceptions.
たべる, to eat, is a RU verb. To turn RU verbs into the て form, it's very easy. All you do is take off the る and add て.
This means that たべる becomes たべて. Easy!
U verbs are not as simple, but with practice it'll become easy to remember.
If the verb ends in う、つ、る (remember, sometimes there are verbs that end in る that just happen to be U verbs and not RU verbs,) then to make it the て form you need to add って.
あう becomes あって。
If the U verb ends in む、ぶ、め then you need んで。
よむ becomes よんで
If the U verb ends in く, then you need to make it いて。
かく becomes かいて。
If the U verb ends in ぐ, then you need to add いで。
およぐ becomes およいで。
If the U verb ends in す then it becomes して。
はなす becomes はなして。
Great! It might take a while to remember all these, but don't worry. When you hear the verbs enough you will start to learn naturally which verb needs which ending.
Also, you need to remember three important exceptions, to go, to come and to do.
いく － いって
する － して
くる － きて
You'll just have to remember these on their own.
I'll be having some more sections with the て form, because now we know how to make it, we need to know what to do with it!