A space is connected if it is not the disjoint union of two nonempty open sets. Connected spaces have no gaps. Continuous images of connected spaces are connected.
Your study of open sets gave you a language for describing topology in terms of neighborhoods and openness rather than distance. Connectedness is one of the first global properties that language can express — it answers the question: is this space "in one piece"? The formal definition is a negation: a topological space X is connected if it cannot be written as X = U ∪ V where U and V are both open, both nonempty, and disjoint. If such a partition exists, X is disconnected — it has been split into two completely separate open pieces with no overlap and nothing between them.
The real line ℝ with its standard topology is connected. Any open interval (a, b) is connected. But consider ℝ minus a single point: ℝ \ {0} = (−∞, 0) ∪ (0, ∞). These two pieces are both open in ℝ, nonempty, and disjoint — a valid disconnection. The intuition is that removing a single point "cuts" the line into two disconnected halves. The integers ℤ with the discrete topology (where every subset is open) are also disconnected: {0} and ℤ \ {0} form a valid disconnection. In the discrete topology every single-point set is both open and closed, and any space with more than one point is immediately disconnected.
One of the most powerful facts about connectedness is its preservation under continuous maps. If f: X → Y is continuous and X is connected, then f(X) — the image — is connected. This theorem has a celebrated corollary you might recognize from calculus: the intermediate value theorem. The argument runs as follows. The real line segment [0, 1] is connected. A continuous function f: [0, 1] → ℝ maps it to a connected subset of ℝ. Connected subsets of ℝ are intervals (a theorem in its own right). So f([0, 1]) is an interval — meaning f takes all intermediate values between f(0) and f(1). The IVT is just connectedness in disguise.
Understanding what makes a space disconnected often matters as much as knowing when it is connected. A space is disconnected precisely when it has a clopen subset — a set that is simultaneously open and closed, other than the empty set and the whole space. In a connected space, the only clopen sets are ∅ and X itself. This characterization is useful for proofs: to show X is connected, assume U is clopen and show it must be ∅ or X. The connected components — the maximal connected subsets of a space — partition every topological space and generalize this idea to spaces that have multiple pieces.