These words are all used when something is appropriate or perfect for a situation. They are also used when talking about clothing.
When two things match, it means they are equal, or very similar so they appear nice together.
I'm buying a yellow hat to match my yellow jacket.
Your socks don't match - one is black and the other is brown.
That modern lamp doesn't match the old-fashioned decor in the bedroom.
I think he's telling the truth; his story matches the evidence.
Whereas match usually refers to color and appearance, fit refers to size. If a piece of clothing fits, it means it is the correct size or shape:
These shoes don't fit me; I need a larger size.
It's hard for me to buy pants because I'm tall - but these jeans fit me perfectly!
The table won't fit through the doorway; we'll need to take it apart to get it inside.
We can also use fit in a more metaphorical sense to mean when somebody belongs in a group; their personality or characteristics are compatible with the others:
Let's hire him - he has a positive attitude and seems like a good fit for the company.
When I was in high school, I loved to study and I felt like I didn't fit in with the popular kids, who were all athletes.
The verb suit, when talking about clothing, is used more generally for when a certain style looks good on somebody:
That green shirt really suits you. It brings out the color of your eyes.
More generally, the verb suit means to be appropriate for. We often use the adjective form suitable:
My new job suits me much better than my old job - I feel like it's more in line with my career goals.
Flip-flops aren't suitable for hiking; you need shoes that will protect your feet.
The comedy movie is more suitable for children than the action movie.