In Linear Algebra, basis is a set of linearly independent vectors $\mathbf v_1, ..., \mathbf v_n$
Vectors $\mathbf v_1, ..., \mathbf v_l$ span a (sub)space $\iff$ this space consists of all possible linear combinations of these vectors
Basis of a vector space is a sequence of vectors $\mathbf v_1, \mathbf v_2, ..., \mathbf v_d$ that
Standard Basis:
1 \\ 0 \\ 0 \end{bmatrix}$, $\mathbf e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, $\mathbf e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$
Non-Example:
1 \\ 1 \\ 2 \end{bmatrix}$, $\begin{bmatrix} 2 \\ 2 \\ 5 \end{bmatrix}$, linearly independent, but don't span $\mathbb R^3$
1 \\ 1 \\ 2 \end{bmatrix}$, $\begin{bmatrix} 2 \\ 2 \\ 5 \end{bmatrix}$, $\begin{bmatrix} 3 \\ 3 \\ 7 \end{bmatrix}$, the 3rd vector is a linear combination of first 2
Another example:
1 \\ 1 \\ 2 \end{bmatrix}$, $\begin{bmatrix} 2 \\ 2 \\ 5 \end{bmatrix}$, $\begin{bmatrix} 3 \\ 1 \\ 8 \end{bmatrix}$
1 & 2 & 3\\ 1 & 2 & 1\\ 2 & 5 & 8 \\ \end{bmatrix}$, rank is 3
Also,
Every space has some basis, and each basis of this space has the same number of vectors. The number of vectors in the basis is the dimension of this space.