The articulation and distribution of /r/ are found in different accents of English and they have many differences. However, one single pronunciation is recommended for learners of English: the post-alveolar approximant.
In order to make the above statement understandable we have to understand the term "approximant". An approximant is an articulation in which the articulators don’t produce a complete consonant (such as a plosive, fricative or nasal) because they don’t get sufficiently close to each other.
Note: Any vowel articulation can be classed as an approximant but the term "approximant" is usually used for consonants.
When we try to produce a long /r/ sound we feel that no part of the tongue is in contact with the roof of the mouth. This is a very important thing because it is very different from the /r/ sound of most of the languages out there, where the tongue-plate contact usually exists.
If we try to produce an alternating sequence of /r/ and /d/ (rdrdrdrdrdrdr) something very interesting we can observe. When /r/ sound is produced, looking in the mirror we can see more of the underside part of the tongue than you can see when producing the /d/ sound. That is because in the case of /r/ sound the tongue tip is raised and a little curled back.
We might be tempted to say that this curling-back process is the same as in case of /t/ and /d/ sounds but producing the /r/ sound takes the tip of the tongue even further back in the mouth. This is the reason why this approximant is called "post-alveolar".
Note: The /r/ sound at the beginning of a syllable, if it follows p, t, or k sound, is voiceless and slightly fricative (a good example is the word "press").
Another interesting characteristic of articulation of /r/ is that when it is produced the lips are slightly rounded. Learners of English, when try to produce this sound (in the correct way), usually exaggerate it producing a sound very similar to /w/ because the lips are too rounded.
One last interesting fact related to /r/ sound is that not all varieties of English pronounce the /r/ sound if it is placed at the end of a word (as in the words "car", "ever" etc) or even if it is the last consonant of the word (as in the word "here").
Many accents of English pronounce /r/ in words like those shown above; most noticeably accent that pronounces /r/ in final position and before a consonant is the American English. These kinds of accents are called "rhotic accents". The accents in which /r/ only occurs before vowels (as in most varieties of British English) are called non-rhotic.