R symbols in math: ℝ, ℜ and ℛ
ℝ (double-struck R) stands for the set of real numbers, as in x ∈ ℝ. ℜ (black-letter R) can denote the real part of a complex number, so ℜ(z) means "the real part of z". ℛ is the script capital R, used for various quantities in math and physics.
All three live in Unicode's Letterlike Symbols block, which is older than the math alphabets. That is why fancy text tools use them in place of the matching math-alphabet letters, which Unicode left empty. See the real numbers symbol page for ℝ and the other number sets.
Circled R vs. ®︎, and other decorative Rs
The registered sign ®︎ is a legal symbol for registered trademarks, while Ⓡ and ⓡ are plain circled letters. The circled letters are fine for initials and usernames, but do not put Ⓡ after a brand name and expect it to work as ®︎. The rules are explained on the registered trademark symbol page.
Bold and script letters like 𝓡 and 𝐑 come from Unicode's math alphabets, so they paste into Instagram, TikTok and Discord names as real text. You can turn whole words into these styles with the fancy text generator.
Backwards R and other R-like symbols
The "backwards R" Я is actually the Cyrillic letter ya. It is popular in faux-Russian lettering and logos, but Russian readers see a different letter, pronounced "ya". The turned ɹ is the phonetic symbol for the English r sound, and ʀ is a small-capital R used in phonetics and small-caps text.
Some symbols are built on an R. ℞ (prescription take) is the Rx sign, from the Latin recipe, "take". ℟ (response) marks the congregation's responses in liturgical books. The Indian rupee sign ₹ blends the Devanagari letter ra with a Latin R.