Get o-rings under reds, browns, or blacks. It stops them from bottoming out loud. If you already have them just put a whole bunch of o-rings under them. Every mech keyboard is still generally loud.

Most brands that say "silent" just refer to using some of the quieter, less clicky switches (reds, blacks, some browns). In reality, most mechanical keyboards are pretty loud by nature, so unless you're getting o-rings underneath your keys to reduce the key-impact it'll be pretty loud.

I'm sure one of the keyboard experts here like smaka could recommend one of the softer ones, but don't fall into the trap that any mechanical keyboard will be quiet.

Also, pretty decent video here detailing the differences between all the switches and what to look for in a keyboard if you're unsure.

https://www.youtube.com/watch?v=ATgetulvk14

As for a non-mechanical keyboard, what would you recommend?

You don't get one.

Nothing wrong with rubber domes. Not everybody can shell out $100+ for a good mech keyboard right now.

Blacks and Reds are going to be the quietest. Don't get blacks; they are a pile of poop. The Razer keyboard is blacks. Black keys are a little bit heavier then Reds so they make your hands tire out faster.

Blacks and Reds are linear switches so they don't offer tactile feedback or a sound response. This is considered ideal for gaming (I'd argue here that Browns are better because of that tactile feedback). They are still louder then Rubber domes be quite a bit since you will probably bottom out your keyboard to press a key.

Browns are a little bit quieter then Blues but not enough to quell the problem you're having. If you have disposable income I'd recommend Topre keyboards (Happy Hacking or Realforce Keyboards). They don't use mechanical switches but use capacitive switches and are significantly quieter and offer the ease and enjoyment that a mechanical keyboard does.

There are quiet mechanical keyboards. They will cost you $200+ because they use Topre switches. Cherry MX-based boards that are advertised as "quiet" or "silent" almost exclusively use Cherry MX Brown switches. The switches are silent. If you type on a mechanical keyboard as it's intended, it will be silent. the problem is that 99.9% of people do not type this way and end up bottoming-out the keys.

Bottoming a key just means that you push the key until it stops moving instead of pushing it only until it actuates. What stops it from moving is the back of the keycap hitting the plate/PCB. This results in a moderate to loud 'clack'. There are only two ways to keep this from occurring. Either stop bottoming-out the keys, or get dampeners as thomas lansky has already pointed out. The problem with the o-rings/other dampeners is that they make the keys feel mushy. It'll always be a mechanical keyboard and the key repeatability and most other benefits will always be there, but you'll essentially lose the solid feel that you get with each undampened keypress. In addition to that, you also lose a certain amount of key travel. Since it prevents you from bottoming out, it shortens the distance that you can press the key before the key hits the dampener. To me, this just feels awkward.

I personally cannot stand using a mechanical with dampeners, but I know of several people who prefer to use them.

http://elitekeyboards.com/products.php?sub=access,slpads

These are the least offensive-feeling dampeners I've used. I would recommend the firm ones as they don't make your keys feel as mushy as the soft ones.

This video should give you a pretty good idea of the difference between dampened and undampened keys.

https://www.youtube.com/watch?v=lFkl1Vet1eU

I'm definitely buying a Realforce some day, those things are beautiful

To clarify a few things from post #7:

1. Topre switches ARE mechanical switches. They are capacitive switches, but they are very much mechanical.

2. If noise is your concern, the type of Cherry MX switch does not matter more than "blue or not blue". Every Cherry MX switch that is not blue is just as loud as the rest. There are certain keyboards that are louder or quieter than others but this is largely based upon whether or not the keys are mounted to a plate or directly on the PCB.