There are no other boundary conditions for (open bosonic) strings: The string is fixed in some directions and free in others. There is one important generalization though:
If you impose Dirichlet or Neumann boundary conditions for each coordinate separately, the ends of the strings will always be confined to affine subspaces of $\mathbb R^n$. But in general, the end of an open string can be confined to any $(p+1)$ dimensional submanifold, called a Dp-Brane. Dp-Branes are extremely important objects in string theory, their significance will probably later become clear in your lecture/book.