What does it mean to have a determinant equal to zero? After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$.

I hope someone can explain this to me in plain English.
For an $n\times n$ matrix, each of the following is equivalent to the condition of the matrix having determinant $0$:


The columns of the matrix are dependent vectors in $\mathbb R^n$
The rows of the matrix are dependent vectors in $\mathbb R^n$
The matrix is not invertible. 
The volume of the parallelepiped determined by the column vectors of the matrix is $0$.
The volume of the parallelepiped determined by the row vectors of the matrix is $0$.
The system of homogenous linear equations represented by the matrix has a non-trivial solution. 
The determinant of the linear transformation determined by the matrix is $0$. 
The free coefficient in the characteristic polynomial of the matrix is $0$. 


Depending on the definition of the determinant you saw, proving each equivalence can be more or less hard.
If a matrix A has an inverse, it means that the matrix multiplication AB = I, the identity matrix.  In plain English then, if a matrix is invertible then it may have a solution. 

If a matrix's determinant is nonzero, the matrix may have a solution. If the determinant is zero, then the matrix is not invertible and thus does not have a solution because one of the rows can be eliminated by matrix substitution of another row in the matrix. 

Common reasons for matrix invertibility are that one or more rows in the matrix is a scalar of the other.

EXAMPLE for a 3x3 matrix: Row 1: x + y + z = P  versus Row 2: Qx + Qy + Qz = QP.  Row two can be rewritten as Q*(x + y + z = P) or Q*(Row 1)]

OR one of the rows can be created by addition of two or more of the other rows.  

EXAMPLE for a 3x3 matrix:  Row 1: x + 2y + 3z = P, Row 2: 4x + 5y + 6z = Q, Row 3: 7x + 8y + 9z = P+Q. You see that Row 3 can be duplicated by adding Row 1 and Row 2.

In short, if the determinant of a matrix is zero, the matrix does not have a solution because the matrix cannot be inverted.