A bivector is an element of Wedge^2 V, so it is dual to a 2-form on V. You can think of a bi-vector as a tiny piece of area.

If V is three dimensional and comes with an inner product, then one can choose an isomorphism between V and Wedge^2 V which commutes with all the orthogonal maps for your inner product. In elementary math, this is the map which we call the cross product. This is not quite unique; you have to decide whether to use the left-hand-rule or the right-hand-rule to take cross products.

In my opinion, the best way to learn to distinguish between vector and bivectors is to get in the habit of not identifying V and V^*. One way to do this is to work with an inner product given by an arbitrary symmetric matrix g and keep the matrix g in all your computations, rather than changing to an orthonormal basis.

A quicker way which I find useful is to think about whether the quantity in question has a natural direction, or has a sign ambiguity which comes from some arbitrary convention. For example, the normal vector to an orientated surface in 3 space is going to be a bivector, because we need to decide whether the orientation circles the normal to the left or the right.

Writing down a bi-vector in d dimensions takes (d choose 2) coordinates. So, for d=4, we need 6 coordinates and we can't fit them into a single vector. I'm guessing that "simple" means a wedge of two vectors. So e_1 wedge e_2 + e_3 wedge e_4 is not simple. Once we get up into higher than 3 dimensions, there is nothing that prevents this, so it can happen.

A bivector is an element of $\bigwedge^2 V$, so it is dual to a $2$-form on $V$. You can think of a bi-vector as a tiny piece of area.

If $V$ is three dimensional and comes with an inner product, then one can choose an isomorphism between $V$ and $\bigwedge^2 V$ which commutes with all the orthogonal maps for your inner product. In elementary math, this is the map which we call the cross product. This is not quite unique; you have to decide whether to use the left-hand-rule or the right-hand-rule to take cross products.

In my opinion, the best way to learn to distinguish between vector and bivectors is to get in the habit of not identifying $V$ and $V^*$. One way to do this is to work with an inner product given by an arbitrary symmetric matrix $g$ and keep the matrix $g$ in all your computations, rather than changing to an orthonormal basis.

A quicker way which I find useful is to think about whether the quantity in question has a natural direction, or has a sign ambiguity which comes from some arbitrary convention. For example, the normal vector to an orientated surface in $3$ space is going to be a bivector, because we need to decide whether the orientation circles the normal to the left or the right.

Writing down a bi-vector in $d$ dimensions takes $\binom{d}{2}$ coordinates. So, for $d = 4$, we need $6$ coordinates and we can't fit them into a single vector. I'm guessing that "simple" means a wedge of two vectors. So $e_1 \wedge e_2 + e_3 \wedge e_4$ is not simple. Once we get up into higher than $3$ dimensions, there is nothing that prevents this, so it can happen.