What ℝ means
ℝ is the set of all real numbers: every point on the number line. That includes whole numbers, negatives, fractions like ½, and irrational numbers such as √2 and π. It does not include imaginary numbers like i, which belong to ℂ.
You will see ℝ in statements like x ∈ ℝ ("x is a real number"), f: ℝ → ℝ (a function from real numbers to real numbers) and ℝ² for the coordinate plane. As an interval, ℝ is (−∞, ∞); the infinity symbol page has more on that.
The blackboard bold number sets
The number systems nest inside each other: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ. ℕ is the natural numbers (whether 0 is included depends on the textbook), ℤ the integers, from the German Zahlen ("numbers"), ℚ the rationals, from "quotient", ℝ the reals and ℂ the complex numbers. The real numbers that are not rational, ℝ ∖ ℚ, are the irrationals.
The double-stroke style is called blackboard bold. It comes from the way teachers write bold letters on a chalkboard, by doubling one stroke. Printed books sometimes use an ordinary bold 𝐑 instead, which means the same thing. Do not confuse ℝ with ℜ (black-letter R), which stands for the real part of a complex number.
How to type ℝ
In LaTeX, write \mathbb{R}. It needs the amssymb or amsfonts package, and the same command works for \mathbb{N}, \mathbb{Z}, \mathbb{Q} and \mathbb{C}. In HTML, use ℝ or ℝ. In Microsoft Word, type 211D and press Alt + X.
There is no keyboard shortcut on Windows or Mac, so copying is the easiest way. A Unicode curiosity: ℝ, ℕ, ℤ, ℚ, ℂ, ℙ and ℍ were encoded early as letterlike symbols, so the rest of the double-struck alphabet, like 𝔸 and 𝕊, lives in a later block that skips those seven letters. Get a full double-struck alphabet with the fancy text generator.