Source: wikipedia
The above animation shows how to interpret the cross product \(\vec a \times \vec b\) of two vectors \(\vec a\) and \(\vec b\):
To figure out which direction the cross product vector goes in, point the index finger of your right hand in direction of \(\vec a\), and middle finger in direction of \(\vec b\). Then your thumb shows the direction of \(\vec a \times \vec b\):
Note that the vectors \(\vec a \times \vec b\) and \(\vec b \times \vec a\) have opposite directions, so the order of the vectors matters in cross product; that is, cross product is not commutative. Cross product is also not associative, which means that you can't put parentheses however you want. For example, let \(\hat i\), \(\hat j\) and \(\hat k\) denote the standard basis vectors. Then, by using the right-hand rule twice, we get $$ \hat i \times (\hat i \times \hat j) = \hat i \times \hat k = -\hat j, $$ but with different parentheses, we get $$ (\hat i \times \hat i) \times \hat j = \vec 0 \times \hat j = \vec 0. $$
By combining this geometrical definition of cross product with dot products and determinants, it's possible to notice that $$ \vec a \cdot (\vec b \times \vec c) = \det \left[ \begin{matrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{matrix} \right], $$ where \(a_x\), \(a_y\) and \(a_z\) denote the components of \(\vec a\), and similarly for other vectors. This leads to $$ \vec b \times \vec c = \det \left[ \begin{matrix} \hat i & \hat j & \hat k \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{matrix} \right], $$ which is the usual way to calculate a cross product. Here the determinant must be calculated as if \(\hat i\), \(\hat j\) and \(\hat k\) were numbers.