Quick answer
Р р is a rolled r, made with the tip of the tongue, like Spanish or Italian. It is never the English r and never the French one.
r
How to say it
Say the American “butter” fast: the tt is one tap of the tongue, which is already a short Russian р. Repeat that tap and let it vibrate.
If you cannot roll it
You will still be understood. An English r in Russian sounds foreign but never unintelligible, and the tap version — one single hit, not a roll — is enough. Most learners get there in a few weeks of trying.
Soft and hard р
Before е, и, ю, я the р is soft: мо́ре, ряд. It is the same roll with the tongue a little further forward.
Typical mistakes
Using the English r in рестора́н
tap or roll the tongue
The English r is made with the back of the tongue and sounds distinctly foreign here.
A French uvular r
tongue tip, not throat
Russian rolls at the front of the mouth.
Check yourself
Which word contains Р?