The span of a set of vectors is the set of linear combinations of those vectors. Let $\{\vec{v_1}, \cdots , \vec{v_p} \} \subset V$. Then
$$ \text{span}(\vec{v_1}, \cdots, \vec{v_p}) = \{\lambda_1\vec{v_1} + \cdots + \lambda_p\vec{v_p}\ \forall \lambda_i \in \mathbb{R}\} \subset V $$
And we say that $\{\vec{v_1}, \cdots , \vec{v_p} \}$ is a spanning set of $V$ if
$$ \text{span}(\vec{v_1}, \cdots, \vec{v_p}) = V $$
Fact: $\text{span}(\vec{v_1}, \cdots, \vec{v_p})$ is a subspace of $V$
$V$ is finite-dimensional if there exists a finite spanning set for $V$. Else, $V$ is infinite-dimensional.
For example, $\mathbb{R}^n, \mathbb{P}_n(\mathbb{R})$ is finite-dimensional. $\mathbb{C}(\mathbb{R}), \mathbb{P}(\mathbb{R})$ are infinite-dimensional.
A set of vectors are either linearly-independent or linearly-dependent. Linear independence implies no redundancy between the vectors, or a zero null-space. By contrast, linear dependence implies redundancy, i.e. at least one vector can be expressed as a linear combination of the others
$$ \{\vec{v_1}, \cdots, \vec{v_p}\} \text{ linearly dependent} \iff \vec{v_j} \in \text{ span}(\vec{v_1}, \cdots, \vec{v_{j-1}}, \vec{v_{j+1}}, \cdots, \vec{v_p}) $$
A basis for a finite-dimensional vector space $V$ is a subset $\{\vec{v_1}, \cdots, \vec{v_p}\} \subset V$ such that
Any two bases of a finite-dimensional vector space $V$ have the same size. The dimension of $V$, $\text{dim}(V)$, is the size of its basis.