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A limit of a diagram (functor) D: J → C is a terminal cone over D: an object L with morphisms to each D(j) compatible with the diagram, such that any other cone factors uniquely through L. Colimits are dual: initial cocones. Limits generalize products, equalizers, and pullbacks; colimits generalize coproducts, coequalizers, and pushouts. A category is complete if it has all small limits, and cocomplete if it has all small colimits; most categories arising in practice (Set, Grp, Top, Ab) are both complete and cocomplete.
Unify previously studied constructions: verify that products are limits over a discrete two-object diagram, equalizers are limits over a diagram with two parallel arrows, and terminal objects are limits over the empty diagram. Dually identify coproducts, coequalizers, and initial objects as colimits.
You have already studied products, equalizers, pullbacks, terminal objects, and their colimit duals. Limits and colimits are the unifying concept behind all of them: they are the right way to say "an object that fits a diagram in the most efficient possible way."
A diagram in a category C is just a functor D: J → C, where J is a small index category encoding the shape of the diagram. For products, J has two objects and no arrows other than identities. For equalizers, J has two objects and two parallel arrows. For pullbacks, J is a cospan shape. The limit of D is a terminal cone over D: an object L together with morphisms L → D(j) for each object j of J, satisfying all the commutativity conditions imposed by J's arrows, and such that any other such cone N → D(j) factors through L via a unique morphism N → L. This unique factorization is the whole content of the universal property — it is not a minimality condition but a *uniqueness* condition.
Colimits are the exact dual. A cocone under D is an object Q with morphisms D(j) → Q compatible with the diagram. The colimit is the initial cocone: every other cocone factors through it uniquely. Coproducts are colimits over discrete two-object diagrams; coequalizers are colimits of two-parallel-arrow diagrams; pushouts are colimits of span diagrams.
Be careful not to confuse categorical limits with analytic limits of sequences. They are conceptually related only in the sense that both describe "convergence to a universal object" — filtered colimits in suitable categories do recover directed limits of sequences, but this is a special case. In general, categorical limits exist in many categories that have no analytic content at all, such as categories of groups or partial orders.
A category is called complete if every small diagram has a limit, and cocomplete if every small diagram has a colimit. Most familiar categories — Set, Ab, Grp, Top, and R-Mod for any ring R — are both complete and cocomplete. This is not automatic, however: the category of finitely generated abelian groups, for instance, fails to have all small limits. Completeness is a genuine structural property, and verifying it is one of the first things you check when working with a new category.