I am taking part in a music theory course on coursera, and I could not understand the difference between a tie and a fermata. Could someone please explain it to me?
A fermata, or pause, lasts an undetermined length of time. Not too long, but at the player's discretion. It can make , say, a semibreve in 4/4 last 5 beats. Ties join the same pitch notes for a couple of reasons. Over a bar line, so the second note doesn't get played again, but carries on; to make up a note length that's not available; and to write notes in a readable manner, such as the second and third beats in 4/4 played as one note. A fermata can also be used over a rest, again, to make it slightly longer than written.
That sounds like a strange music theory course since a tie and a fermata are absurdly different things.
A tie means that a note has to be written as several consecutive notes for notational reasons (interfering bar lines or line breaks, hard to read rhythm, unrepresentable total note duration) but is a single note with the total length of all tied written notes.
A fermata means that a note serves as a temporary (or actual) ending and should be held for longer than its nominal value: the music is coming to a stop here. Usually fermata are on notes ending at the same time in all instruments but there are cases where a new start threads into a fermata, naturally in a voice that cannot itself end in a fermata.
There also is the "laissez-vibrer tie": those are ties ending in the void. They indicate for percussive instruments like guitar to not dampen (or unfinger) the strings in question after the preceding note has ended but just let them ring out naturally while continuing without pause otherwise.
But while a fermata means "end this note late, stopping the clock", "laissez-vibrer" means to not give any determinable ending and keeping the speed.