Is this a brass or copper elbow pipe under the sink?
It is Brass.

You will probable not find that elbow at an all-round home improvement center, you will need to go to a dedicated plumbing supply house. ( take the elbow with you to insure you get a match ).   

The "tightening sleeve rings" are called slip joint nuts, it  is likely that yours are still fine and reusable.

Because  the top of your  elbow is connected to the drain basket it is acting like a tail piece. A tail piece to basket connection is not a slip joint, it uses a flat washer instead of a slip joint washer but it does use the same nut. For this reason it may be difficult to find the correct elbow.

If you do decide to replace it all with plastic you will need to know what the interface is with the Sanitary tee in the wall. It appears to be a threaded connection and a plastic set up would have a slip joint connection, even if it is threaded you should still be able to slip the 1 1/2 plastic pipe into it but investigate it before buying parts. 

Also you will want a  dishwasher branch tail piece . It appears you do not have an air gap for your dishwasher but it is best to have one, it may be code in your area, if not put in a high loop.( explained in this stack question )