What is the physical meaning of a dot product and a cross product of vectors? My teacher told me that Vectors are quantities that behave like Displacements. Seen this way, the triangle law of vector addition simply means that to reach point C from point A, going from A to B & then to C is equivalent to going from A to C directly.
But what is the meaning of a product of vectors? I cannot imagine how a product of displacements would look like in reality. Also, how do we know whether we need the scalar (dot) product or a vector (cross) product?
 A: You seem to look for geometrical meanings. The cross product gives the area of the parallelogram that is spanned by the two vectors as the length of the resulting vector and the direction perpendicular to both vectors.
The scalar product gives you information about the component of one vector into the direction of the other.
As Georg said, you will probably know when you need it. I also found that school is making this stuff more complicated than it needs to be by just letting the students memorize particles of information instead of teaching understanding.
If you have to stay with memorizing, a pretty clear way for distinguishing scalar and vector product is the result in respekt of the direction of the vectors: the cross product gives the maximal value, if the vectors have a 90° angle between each other and 0 for 0°, the scalar product.
About the meaning: I would not think in displacements. A force has nothing to with displacements for starters. A vector is a scalar quantity with a direction, or even more general, just a bunch of numbers - generally more than 1 - with a certain operation like the possibility to add two vectors.
A: It is a bit misleading to think of Vectors as displacements. Vectors are abstract mathematical objects that live in a Vector Space over a Field (say Real number field). A vector is a higher-order animal that is obtained when you pour the Field over a Vector group. 
Quick and dirty introduction:


*

*Build a set with a collection of objects.

*Establish a relationship between the objects by means of an operation. (Say multiplication).

*See if it forms a Group. (We assume yes).

*Now bring in a Field (Set which forms a Group under two operations) and form a new algebraic structure called a Vector Space over a Field by establishing certain combination rules between elements of the Field and the elements of the group. To make life easier we choose a field that has one operation the same as the Group.


The Field serves to fill the "holes" between elements in the Group by giving you the ability to scale vectors. Vector "Products" are obtained by asking the question "how do we make vectors talk to each other"? Inner products yield elements in the Field (scalars) and wedge products yield another vector that is not in the same sub-space as the two original vectors.
How do you know whether a physical system can be represented by an inner or outer product?
Well, the easiest way to check is experimentally. For example how do we know if $\vec{F}=q(\vec{v}\times\vec{B})$ and not $q(\vec{B}\times\vec{v})$ ?? This is by experiment.
*Remember that when we measure something, we do so in the Field because our results are numbers.*This is a critical concept. 
There is a lot more to say and I'll edit this when I have the time. Abstract Algebra is a beautiful subject. Hope this helps. :)
Edit #1:
The triangle law of addition comes out naturally when you write down the rules that result in the formation of a vector space. All these geometric pictures are misleading because they are presented to students as the absolute concept. You can ask the question "Why is a vector represented by an arrow?". My opinion (I have never seen this discussed anywhere) is that by giving a "direction" you inherently establish an ordering within the set. A lot more can be said if you think deeper, but I guess I have confused the OP already. :) :)
