How big should my Time Machine backup drive be? My 13" MacBook Pro has 500GB internal hard drive space, and I bought a 1TB external HDD to use as my Time Machine backup. I only read from many articles that the backup should be about 2 to 3 times the size of my hard drive, but I feel like 1TB will never be filled up, and in fact, leave a lot of free space.

I understand that Time Machine takes a snapshot of my whole system and stores it in the backup drive. But say, for example, only about 120GB of my internal HDD is in use. Then it'll only take a snapshot that's about 120GB in size, and while that same snapshot will grow over time as I take more recent snapshots of my system, it will never go beyond 500GB because that's the maximum size of my internal HDD. Or do I have it wrong? How does Time Machine work, really?

I'm asking this because I want to know if I should use a separate external HDD for storing my actual files.
You probably want to check out this website for the most information, but here is the answer to your question, quoted from the website above:
1. How big a drive do I need for Time Machine?

A general &quot;rule of thumb&quot; is, to keep a reasonable &quot;depth&quot; of backups,
Time Machine needs 2 to 4 times as much space as the data it's
backing-up (not necessarily the entire size of your internal HD).  Be
sure to add the size of the data on any other drives/partitions you
want to back up.
But this varies greatly, depending on how you use your Mac.  If you
frequently add/update lots of large files, then even 5 times may not
be enough.  If you're a light user, you might be able to get 1.5 times
to work, but that's subject to problems any time a large backup is
needed.
And, of course, the larger the drive, the more old backups Time
Machine can keep for you.  A drive that's too small may only have room
for a few weeks (or even days) of backups.
Unfortunately, it's rather hard to predict, and most of us have a
tendency to add more and more data to our systems over time, so if in
doubt, get a bigger one than you think you need now.
Also, there are some OSX features and 3rd-party applications that take
up large amounts of backup space, for various reasons.  See question
9  for details.
This is a trade-off between space and how long Time Machine can keep
its backups, since it will, by design, eventually use all the space
available.  But it won't just quit backing-up when it runs out:  it
starts deleting the oldest backups so it can keep making new ones.
Thus, the more space it has, the longer it can keep your backups.
If your backup disk is on the small side, and Time Machine needs to do
a very large backup, either because you've added or changed a lot or
done something like an OSX update since the previous backup, you may
get one of the messages in Troubleshooting item #C4  (which one
depends on exactly what happened, and which version of OSX you're on.)

My Answer

Then it'll only take a snapshot that's about 120GB in size, and while that same snapshot will grow over time as I take more recent snapshots of my system, it will never go beyond 500GB because that's the maximum size of my internal HDD.

No, not quite. The 1 TB hard drive will be filled up because Time Machine keeps your backups and deletes them once your hard drive is filled. There is more than one backup stored on that hard drive. As stuffe pointed out as well, Time Machine allows you to restore to a previous backup, since Time Machine may keep 7 or 8 backups on that 1 TB hard drive.

I'm asking this because I want to know if I should use a separate external HDD for storing my actual files.

Yes, you should. You should be dedicating a drive to Time Machine. Here is the quoted answer from 3 in the website I linked above:
3. Can I use my Time Machine disk for other stuff?

Yes.  Time Machine will not delete anything you put there.  But it's
not a good idea to put anything else important on the same physical
drive, unless you back it up elsewhere.  When (not if) that drive
fails, you risk losing it.
If you want to do this anyway, it's much, much better to partition an
external drive into 2 (or more) parts, also called volumes.  Assign
one to Time Machine, for its exclusive use for backups;  use the other
partition(s) however you want.  To use a new drive, or one you don't
mind erasing, see question #5.  To add a partition to an existing
drive that already has data on it, see question #6.