Regularly cleaning the saxophone mouthpiece is an essential part of instrument care.
Think about it: how hygienic would it be to not brush your teeth?
Here's an incredibly simple (and highly effective) way to deep clean your mouthpiece for years to come:
Step 1: add room-temperature water to a small bowl
NO HOT WATER! That will ruin the outer coating of the mouthpiece, which causes discoloration
Step 2: pour 1 tablespoon of white vinegar
Nothing fancy here: just plain, cheap, boring, store-brand white vinegar. Don't be tempted to add extra vinegar for an especially dirty mouthpiece: the key here is time, not quantity.
Step 3: soak for 15 minutes
I start with 15 minutes and check to see if the vinegar has worked its magic. If the white build-up comes off easily, the mouthpiece is ready. If it doesn't, continue to soak in 15 minute increments until it does.
If you haven't cleaned your mouthpiece in a long time, your soaking period may be extended quite a bit. If you regularly clean your mouthpiece (say, every 2-3 weeks), 15 minutes will be more than sufficient
Step 4: use a Q-tip to gently remove any white spots
NEVER use a mouthpiece brush! Those can damage and scratch the interior and exterior of the mouthpiece, which is an expensive mistake
Step 5: rinse with cool water and pat dry