The Borel sigma-algebra is the sigma-algebra generated by open intervals on ℝ (or open sets in ℝⁿ), containing all open sets, closed sets, and their countable unions and intersections. It is the natural sigma-algebra for probability on ℝ and avoids pathological non-measurable sets. Essentially all 'reasonable' subsets are Borel measurable.
You know two things coming in: what a sigma-algebra is (a collection of subsets closed under complement and countable unions), and what open sets on the real line look like (unions of open intervals like (0, 1) and (2, 5)). The Borel sigma-algebra ℬ(ℝ) is defined simply as the smallest sigma-algebra on ℝ that contains all open sets. Everything else follows from unpacking what "smallest sigma-algebra containing a collection" means.
The construction is universal: given any collection 𝒞 of sets, the sigma-algebra *generated by* 𝒞 is the intersection of all sigma-algebras that contain 𝒞. There is always at least one such sigma-algebra (the power set of ℝ contains every subset), so the intersection is well-defined and is itself a sigma-algebra. The Borel sets are generated by the open intervals, but you could equivalently use closed intervals, half-open intervals [a, b), or even just the rays (-∞, x] — they all generate the same sigma-algebra. This robustness is one sign that ℬ(ℝ) is capturing something natural about the real line.
What does ℬ(ℝ) actually contain? Open sets are Borel by definition. Closed sets are Borel (as complements of open sets). Countable intersections of open sets — called G_δ sets — are Borel. Countable unions of closed sets — called F_σ sets — are Borel. Continuing this hierarchy transfinitely, the Borel sets form an enormous collection. The remarkable practical fact is that every set you can explicitly describe using a rule or formula is Borel. Non-Borel sets exist (the Vitali set is the standard example) but require the Axiom of Choice for their construction — they are genuinely pathological and impossible to write down concretely.
The reason this matters for probability is that you want statements like P(X ≤ 3), P(X ∈ (1, 2)), and P(X is rational) to have well-defined probabilities. For any of these, you need the corresponding subset of ℝ to be measurable. The Borel sigma-algebra is precisely the right choice: it is large enough to contain every set you would ever naturally ask about, but defined tightly enough to avoid the measure-theoretic paradoxes that arise from the power set. When you define a random variable as a measurable function from (Ω, ℱ, P) to ℝ, the sigma-algebra you equip ℝ with is always ℬ(ℝ) — this is what makes probability statements about the values of X well-defined.