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For an ideal I of a ring R, the quotient ring R/I consists of cosets a + I with addition and multiplication defined component-wise. The natural map R → R/I is a homomorphism with kernel I.
From your study of ideals, you know that an ideal I ⊆ R is a subring that "absorbs" multiplication from R: if a ∈ I and r ∈ R, then ra ∈ I. The quotient ring R/I is the construction that forces every element of I to become zero — by declaring that two ring elements are "the same" whenever their difference is in I.
The elements of R/I are cosets a + I = {a + x : x ∈ I}. Two elements a and b represent the same coset if and only if a − b ∈ I. We add and multiply cosets by choosing representatives: (a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = (ab) + I. That this is well-defined — that the result doesn't depend on which representatives we chose — is exactly what the ideal condition guarantees. Without the absorption property, multiplying by different representatives could yield different cosets, breaking the structure.
A canonical example: take R = ℤ and I = (n), the multiples of n. Then ℤ/(n) is exactly ℤ/nℤ — ordinary modular arithmetic. Computing "5 × 7 mod 12" is working in ℤ/(12). A more algebraic example: take R = ℝ[x] and I = (x² + 1). In ℝ[x]/(x² + 1), the element x satisfies x² + 1 = 0, i.e., x² = −1. This quotient ring is isomorphic to ℂ — the construction adjoins a square root of −1 by "setting the polynomial x² + 1 equal to zero." Quotient rings are the precise algebraic mechanism for enforcing polynomial relations.
The natural map φ: R → R/I sending a ↦ a + I is a surjective ring homomorphism, and its kernel is exactly I. Every ideal is the kernel of some homomorphism, and every kernel is an ideal — these concepts are two sides of the same coin. The First Isomorphism Theorem for rings, which builds directly on this, makes it precise: if φ: R → S is a surjective ring homomorphism with kernel K, then R/K ≅ S. The quotient construction is thus the universal way to build a ring in which a given ideal has been collapsed to zero.