Small uncountable cardinals or cardinal characteristics of continuum are various cardinals which are typically between $\aleph_1$ and $2^{\aleph_0}$ and their definition often has a combinatorial flavor. Some examples are:

• The cardinal $\mathfrak p$ - the smallest cardinality of subsystem of $[\omega]^\omega$ with strong finite intersection property and no pseudointersection.
• Various cardinals related to $(\omega^\omega,\le^*)$ such as the bounding number $\mathfrak b$ (=the smallest cardinality of an unbounded subset) or the dominating number $\mathfrak d$ (=the smallest cardinality of dominating subset).

See also: Cardinal characteristic of the continuum on Wikipedia.