You want a rug that does not have a rubber backing. There are non skid mats, which are sold separately, for under rugs that are safe for most hardwood floors (check the label), but most rubber backed rugs are not. I have an ordinary rug I picked up at Menards for not much in front of my sink:)

If you want something for under a table or a runner, I recently bought an indoor outdoor rug and runner for my living room - it is actually pretty attractive, though not cushy. I needed something easy to clean because I have two dogs. This can be hosed off outside if needed. It also was not too expensive - 8 x 10' for about $100 and I think the 2 x 10' runner was about $75. I bought the rug at a local rug / flooring store, but bought the matching runner online from Overstock.

You can do a search on Houzz for "indoor outdoor rug" under shopping to see some examples.

There are cushier rugs, but they may need special cleaning. Some people have special matts (anti fatigue matts) in front of the sink that are supposed to help tired feet - something I should look into myself.

I agree about a rug you can wash. I bought a cotton runner online that I can throw in my washer. It's not cushy, so it depends on what the rug is for. I spill, so I can't imagine using some of the "antique persian rugs" shown in Houzz photos.

Overstock is a good source for rugs.