The simplest way to show how to multiply a matrix and a vector is by an example. If your browser doesn't handle animated GIF's, or if the download time is too long, I hope to have a link established here eventually to explain it in a Lynx-friendly way.
The shape of the matrix and the vector have to agree. If you have a 10 (row) by 12 (column) matrix you can right-multiply it by a 12-element vector, and your answer will be a 10-element vector.
If A is a 10 by 12 matrix, and B is a 12 by 42 matrix, you can multiply A B to get a new matrix C which is 10 by 42. You cannot multiply B A , since the matrix ranks don't match.
You can think of the multiplication this way: Take the column vector on the right and turn it sideways above the matrix. Now take the dot product of that with each row in the matrix in turn. The dot product value is the answer for that row in the column vector.