Discrete distribution most frequently refers to a distribution that only achieves integer values, and more generally a discrete distribution defines probabilities for distinct potential outcomes that do not cover any continuous interval range. For example, a Poisson distribution often is used to model the number of radioactive decays detected, and those can be any natural whole number including zero counts.
Finite support refers to the range of values that a distribution can achieve. For example, two dice can have an summed outcome from 2 to 12 of the number of dots face up on those thrown dice. We can symbolize this as some integer value $i$ on the interval $[2,12]$. Those dice outcomes are also discrete outcomes. Thus, the distribution of outcomes is discrete with finite support.
Armed with this information, take a look again at the Wikipedia entry. That entry also details other distributions and other support. For example, other distributions can have semi-infinite support, e.g., on $[0,\infty)$ and be discrete (Poisson) or or continuous (Gamma distribution).