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A subring S is closed under both operations. An ideal I satisfies rI ⊆ I and Ir ⊆ I for all r ∈ R, making it a kernels of homomorphisms. Principal ideals generated by single elements are most tractable and central to ring theory.
You already know that a ring has two operations — addition (forming a commutative group) and multiplication (associative, distributing over addition). A subring is a subset that is itself a ring under the same operations: it must contain 0, be closed under addition and subtraction, and be closed under multiplication. Think of ℤ sitting inside ℚ: integers are closed under addition, subtraction, and multiplication, so ℤ is a subring of ℚ. Note that a subring doesn't need to contain 1 (multiplicative identity), though many authors require it — check your definition.
An ideal is a strictly stronger structure than a subring. A subset I ⊆ R is an ideal if it's closed under addition and subtraction (making it a subgroup of (R, +)), and if multiplying *any* element of R by *any* element of I stays inside I: for all r ∈ R and a ∈ I, both ra ∈ I and ar ∈ I. The key example: the even integers 2ℤ = {..., −4, −2, 0, 2, 4, ...} inside ℤ. Any integer times an even integer is even — so 2ℤ is closed under multiplication by all of ℤ, not just by other even integers. This "absorbing" property is what separates ideals from subrings.
The most tractable ideals are principal ideals: ⟨a⟩ = {ra : r ∈ R}, all multiples of a single element. In ℤ, ⟨6⟩ = {0, ±6, ±12, ...} is all multiples of 6. In ℤ, every ideal is principal — there's no way to have an ideal that isn't generated by a single integer (it's generated by the smallest positive element it contains). Rings with this property are called principal ideal domains (PIDs); ℤ is the canonical example. Not every ring is a PID, and identifying when ideals are or aren't principal is a central theme in ring theory.
The reason ideals matter structurally is that they are exactly the kernels of ring homomorphisms — just as normal subgroups are kernels of group homomorphisms. Given an ideal I ⊆ R, you can form the quotient ring R/I, where two elements are identified if their difference lies in I. This is modular arithmetic in disguise: ℤ/nℤ is the ring of integers mod n, where ⟨n⟩ is the ideal of multiples of n. Ideals classify how a ring breaks into equivalence classes, and studying maximal and prime ideals reveals the ring's deep algebraic structure.