Am I correct in assuming that a circuit is given a breaker based on the gauge of the wire installed? e.g. 15 amp circuit has 14 gauge, 20 amp 12 gauge etc.

By way of follow up, therefore if the load on a 15 amp circuit keeps tripping the breaker then either the circuit must be split up or the entire thing must be rewired in order to make it a 20 amp?

For a first order approximation, that's true. The CB protects the wiring and the load.
You can have 12 AWG wire and a 15 A or 20 A breaker, but you can only have a 15A breaker for 14 AWG wire.

Circuits are determined my loads and if they are on more than 3 hrs in a day and circuits are also determined by location.

e.g. Bathrooms require GFCI's and at least a 20 Amp circuit.
Location = bathroom
Bathroom = GFCI
Bathroom = 20 Amp circuit
Because of hair dryers, curling irons and water

An electric water heater
1000 W, tankless, 120 V, 240V, 2 services of 240 V?
In this case, the dedicated circuit's load determines the breaker size and even how far away it is from the panel.

Whether your house is supplied with 100 A, 150 A, 200A or 400 A service depends on things like how many buildings, pool? Hot tub, Guest house, electric stove, electric heat, woodshop etc.

Your followup:

If a circuit keeps tripping, you:

1. Must add up the loads in VA, Watts etc.
2. Determine which are on more than 3 hrs at a time. Multiply those values by 1.25
3. Determine the loads that are on < 3 hrs at a time.
4. Sum 2 & 3
3. Divide by the voltage to get the current
4. Size to next size breaker
5. This is the breaker you need
6. Wire size and distance factors figured in.

7. Determine if it's plausible that something is wrong. Fix, repair, upgrade.

Any new changes should be done to new code requirements unless there are exceptions.

Here's sort of an example:

When you use the microwave and the toaster at the same time, the breaker pops or when you use the microwave and the sweeper at the same time the breaker pops.

Solution: Don't do these things
Upgrade microwave to dedicated circuit.

Excellent, thank you.

That explains why the wife's blow dryer keeps tripping the CB; it's only a 15A circuit. Plus it's has an addition room on it.

Two other questions How can I tell what size service I have in the house, 100 AMP, 150, etc,

Second, (I'll probably get busted in the chops for this, but) How do I know how many breakers, really how much circuit breaker amperage, I can put on this service; just fill up the box?

1. The size of the main breaker. If is a double-breaker with a 100 on it you have 100 A service. To find out if the wiring will support more is another problem altogether.

2. There is a number marked on the breaker box called "maximum number of poles".
This is essentially the number of wires connected to breakers. It can exceed the number of spaces. There are twin breakers (with 2 different or same ratings) that allow two circuits in a single space.

You don't add the breaker capacities.

3. When you exceed capacity of the number of breakers a sub-panel is usually added or the service is upgraded.