A few examples of what you can ask Wolfram|Alpha about:
solve a polynomial equation
solve a system of linear equations
solve an equation with parameters
compute properties of a polynomial in several variables
factor a polynomial
compute properties of a rational function
compute a partial fraction decomposition
compute properties of a vector
compute a cross product
do basic arithmetic on matrices
compute eigenvalues and eigenvectors of a matrix
get information about a quaternion
do calculations with quaternions
get information about a finite group
ask about a property of a group
do algebra with permutations
compute properties of a finite field
compute a specific property
compute the domain of a function
compute the range of a function