To add to this node is to invite disagreement from speakers of unrepresented accents, so the following is only the most cursory of clarifications.

When raising the tongue "in the direction of" the palate, there are two slightly different places you can aim for. Between the teeth and the palate is a small ridge called the alveolar ridge. If you position the tip of the tongue just behind that, you are making a postalveolar sound; if you curl it back somewhat further you are making a retroflex sound.

You can just leave it there and make a continuous sound (technically a continuant) or you can tap it once against the bit of roof of mouth you're aiming at. If you tap it repeatedly fast, it's called a rolled R.

With these parameters we can examine the regional possibilities:

  1. Most of England, Wales, South Africa, Australia, New Zealand: it is a post-alveolar continuant when followed by a vowel; otherwise silent.
    • so it's pronounced in: red, bread, barrow
    • and in the expression: 'war and peace'
    • and silent in: war, water, hard
  2. Ireland: it is a post-alveolar continuant whenever written.
  3. Scotland: it is an alveolar tap whenever written.
  4. South-Western England: it is a retroflex continuant whenever written.
  5. America generally: it is a retroflex continuant when not preceded by a vowel; when a vowel does precede, the retroflexing of the tongue occurs throughout it.
    • so it's a consonant in: red
    • but actually part of the vowel in: fur, war
Accents 2. to 5. are called rhotic; accents of type 1. are non-rhotic (pronounce R in only some positions). Some American accents in New England and in the South are non-rhotic.

An alternative pronunciation in some American accents is what's described as a bunched articulation. I'm not clear about what this is, as I can't produce the sound convincingly to myself or "hear" two American R sounds, but it is widely used. American also has a tap, but it is how the letter T between vowels is pronounced.

No accent actually has a rolled R, though caricaturists use it for Scottish. (Possibly the Scottish R was formerly rolled; but in everyday speech now it's a tap.)

Certain parts of northern England have a uvular sound, made in the back of the throat.

In the modern London/S.E. England variety called Estuary English, the R sound may be more like a W. The consonant is pronounced or silent according to rule 1, but when it's pronounced it's a W-like sound, though not exactly the same as the ordinary W of wide, white. So ride, right aren't exactly the same as these, but very close. Phoneticians I've discussed this with are unclear on the exact nature of the new RW sound. (In fact I think it might be the normal [r] with advanced tongue root.) It is very common among younger speakers in London.

A similar but not identical feature occurred in older varieties of Received Pronunciation, but only among a very few educated and aristocratic types, such as the politician Roy Jenkins ("Woy") and the aged Archdean in The Princess Bride ("mawidge"). (In technical terms, I have seen this described as a rounded uvular approximant.)