According to my research, my roof is what they call a Built-Up Roof.
It is built layer after layer. A total of 5 layers if I remember
correctly
from the roofer.

Layers of what? Shingles laid on top of old shingles?

I am not sure what flashing means

Stop. Call a roofer.

I have plenty of buckets here. So, it should not be any more
problem here if it keeps rainning.

Catching the water in buckets after it's already ruined your ceiling joists,
drywall, insulation and other parts is an incredibly stupid plan.

Call a roofer, get it done right before you end up wrecking the entire
thing.