Two answers.I'm thinking about trees because of the two out of three property: For a simple (no multiple edges) undirected graph G, any 2 of the three conditions

1. cycle-free (better acyclic)
2. connected
3. #edges=#vertices-1

Means G is a tree.

Of course that isn't what you asked. Condition three is not one word although we could coin uni-deficicient

so simple+undirected+acyclic+connected defines tree.

Certain sets are relations (basically any set of ordered pairs). Not counting that as a condition

For a relation, reflexive+symmetric+transitive defines Equivalence relation

similarly reflexive+antisymmetric+transitive defines partial order

Actually there are $k$-ary relations for other $k$ so one could quantify over all relations and restrict to binary relations.