What is the difference between a finite verb and a modal verb?

A finite verb is basically the most normal kind of verb that you are used to working with.  It functions with a subject and shows tense.  That means it can be used in the past, present, or future tenses.  

Let's use the word "type" as an example.  

Past: I typed this response.

Present: I type this response.  

Future: I will type this response.  

Modal verbs are very unique verbs.  They have their own set of "weirdness" to them.  For example, they never take an "s" in the third person.  To make a modal verb negative, you add the word "not" to them.  Lastly, most modal verbs cannot be used in the past or future tenses.  

Some common modal verbs are "should," "would," and "could."  

Some example sentences:

"He should be home shortly."  The third person doesn't take an "s." 

"He could not hit the target."  Adds "not" to make it negative. 

"He will would be with us."  Not correct.