Vector help...

Im trying to understand tangent space and therefore apply it in Blitz... I'm not very good with vectors though, can someone explain some of this..?

//X being the cross product
For each face {
	(x,y,z) = (p2->x-p1->x,p2->u-p1->u,p2->v-p1->v) X (p3->x-p1->x,p3->u-p1->u,p3->v-p1->v)
	if (x!=0) {
		NORMALIZE(x,y,z)
		p1->Tx += -y/x;
		p1->Bx += -z/x;
		p2->Tx += -y/x;
		p2->Bx += -z/x;
		p3->Tx += -y/x;
		p3->Bx += -z/x;
	}
	(x,y,z) = (p2->y-p1->y,p2->u-p1->u,p2->v-p1->v) X (p3->y-p1->y,p3->u-p1->u,p3->v-p1->v)
	if (x!=0) {
		NORMALIZE(x,y,z)
		p1->Ty += -y/x;
		p1->By += -z/x;
		p2->Ty += -y/x;
		p2->By += -z/x;
		p3->Ty += -y/x;
		p3->By += -z/x;
	}
	(x,y,z) = (p2->z-p1->z,p2->u-p1->u,p2->v-p1->v) X (p3->z-p1->z,p3->u-p1->u,p3->v-p1->v)
	if (x!=0) {
		NORMALIZE(x,y,z)
		p1->Tz += -y/x;
		p1->Bz += -z/x;
		p2->Tz += -y/x;
		p2->Bz += -z/x;
		p3->Tz += -y/x;
		p3->Bz += -z/x;
	}
}
For each vertex {
	NORMALIZE(T)
	NORMALIZE(B)
	newN = TxB
	//Check direction of N according to a classic normal computation method:
	//sometimes N needs to be reversed
	if newN . oldN > 0
		N = newN
	else	N = -newN
}

I don't understand how to normalize a vecotr (alothugh i heard its meant to result in the length being 1?) and what does it mean 'X is the cross product.' The cross product of what? Just the vertex? :) If anyone can go through that code, i'd really appriate it xD