On my polished wood table any disturbance (and I have kids, so you can imagine the disturbances...) can shift tiles around, which I then feel compelled to try to adjust to be flush with those around them. I want to get a mat to fix this, and was wondering if anyone has had experience doing this, and can offer advice. Thanks!
You could try some no-slip shelf liner (the kind you use for kitchen cabinets). They're sold by the roll at various department stores, fairly cheaply. They lay flat and can be cut into whatever shape you need. If you can, I would splurge and err on the side of using a heavier-duty liner. The cheap stuff is probably little better than a tablecloth.
For Carcassonne specifically, you'll want to make sure you cover as much of the table as is practical. You probably don't want to artificially constrain the game if your no-slip surface isn't big enough.
Felt. A thousand casinos can't be wrong. Padded felt also makes it easier to pick cards/chits up off the table.
For Settlers of Catan, there is an expansion you can get that consists of six interlocking long sections that frame the pieces in. At worst, your entire island will shift, instead of individual hexes. I just picked it up recently from my local game store after seeing our island moved one too many times.
For other tile-based games, I would suggest a tablecloth like Egg of P'an Ku suggested, or a large piece of fabric (or felt) that you can get at a craft store -- you don't want any fabric to be TOO slippery, though. You can even go one step further and get those no-slip rubber mats for rugs to place between the table and the tablecloth. The only downside to that I could see is the tablecloth may occasionally get bunched up.
You could also try a table-top gaming mat, or simply skip the tablecloth and just use a large number of the no-slip mats I mentioned above, and place your tiles directly on the rubber mats.
It was quite a bit of work to construct, but we made a mat to play games on that was baize on one side and leatherette on the other. It's really nice playing card and board games on the baize surface - it's easy to pick up cards and tokens, as if from a slightly squishy card table. The leatherette underside is heavy enough to keep it in place, and easy to clean if you discover afterwards that the table wasn't as clean as you thought. I made an instructable showing how we made this.
It'd be nice to know of places that sell something like this.
For Settlers of Catan, a sheet of acrylic or pane of glass over the board can work wonders!
I have used both a pane of glass and a sheet of plexiglass. The two are roughly equivalent to play on. Pieces slide around a little bit easier, but not so much as to become a nuisance. I prefer glass for its superior durability, but I have been using plexiglass at home for quite a while and it works just fine.
I just took one of my old Bridge tables and is using that for most games, it works great as long as we're 4 players. I also have a fairly cheap 8-sided poker foldable tabletop thing for bigger sessions (with the added bonus of drinkholders).
This means another table which is perhaps not an option for you but both options can be folded and take very little space when not playing at least.
In addition to Kristo's answer, if you have a wood shop you likely have a router mat which would accomplish something similar.