Cross products are often used with pseudovectors (aka axial vectors). Less with vectors (aka polar vectors). Understanding the difference between axial and polar vectors helps here.

Both axial and polar vectors are what mathematicians would consider a vector. Both are a set of 3 coordinates. They are often drawn as arrows. They can be added together and multiplied by numbers like arrows.

Physicists require something more to consider a quantity to be a vector. They must represent a physical quantity that transforms in the right way when you change the basis.

Polar vectors represent quantities like distance, velocity, acceleration, and force. These can describe motion of a point particle with a magnitude and direction.

Axial vectors represent a different set of quantities, like angular velocity and angular momentum. These describe things like rotary motion in a plane. They are a magnitude and orientation of the plane. This is equivalent to motion around an axis. They are often represented by an arrow, where the arrow is parallel to the axis and perpendicular to the plane. Plane orientation include the idea of clockwise vs counter clockwise. This is represented by putting the arrow on one side or the other of the plane as dictated by the right hand rule.

Axial vectors often arise as the product of two perpendicular polar vectors. $\omega = r \times v$.

For a rigid object fixed to an axis, each point can only move with $v$ perpendicular to $r$. But a free particle can move an any direction. For this case, the cross product picks out the component of $v$ that is perpendicular to $r$, the component that contributes to rotation around the axis. The result is an vector perpendicular to $v$ and $r$ in accordance to the right hand rule.

Magnetic field is an axial vector. See Why is the B-Field an axial Vector? for more. This means a current generates a $B$ field around it, described by magnetic field lines. For a straight line current, the field lines are planar and circular. For more complex currents, they are always closed curves. At any point, the field line is the "axis" that is perpendicular to the plane of the magnetic field.

Magnetic force is generated when a charge moves in the plane of $B$. That is, when a charge moves perpendicular to the "axis" of B. This is captured by $F = qv \times B$.

Cross products are often used with pseudovectors (aka axial vectors). Less with vectors (aka polar vectors). Understanding the difference between axial and polar vectors helps here.

Both axial and polar vectors are what mathematicians would consider a vector. Both are a set of 3 coordinates. They are often drawn as arrows. They can be added together and multiplied by numbers like arrows.

Physicists require something more to consider a quantity to be a vector. They must represent a physical quantity that transforms in the right way when you change the basis.

Polar vectors represent quantities like distance, velocity, acceleration, and force. These can describe motion of a point particle with a magnitude and direction.

Axial vectors represent a different set of quantities, like angular velocity and angular momentum. These describe things like rotary motion in a plane. They are a magnitude and orientation of the plane. This is equivalent to motion around an axis. They are often represented by an arrow, where the arrow is parallel to the axis and perpendicular to the plane. Plane orientation include the idea of clockwise vs counter clockwise. This is represented by putting the arrow on one side or the other of the plane as dictated by the right hand rule.

Axial vectors often arise as the product of two perpendicular polar vectors. $\vec\omega = (\vec r \times \vec v)/r^2$.

For a rigid object fixed to an axis, each point can only move with $v$ perpendicular to $r$. But a free particle can move an any direction. For this case, the cross product picks out the component of $v$ that is perpendicular to $r$, the component that contributes to rotation around the axis. The result is an vector perpendicular to $v$ and $r$ in accordance to the right hand rule.

Magnetic field is an axial vector. See Why is the B-Field an axial Vector? for more. This means a current generates a $B$ field around it, described by magnetic field lines. For a straight line current, the field lines are planar and circular. For more complex currents, they are always closed curves. At any point, the field line is the "axis" that is perpendicular to the plane of the magnetic field.

Magnetic force is generated when a charge moves in the plane of $B$. That is, when a charge moves perpendicular to the "axis" of B. This is captured by $\vec F = q\vec v \times \vec B$.