Typically an activating mutation on one strand can constitutively activate an oncogene which will drive positive selection of the cancer.  Additional mechanisms of oncogene activation can include amplification of its DNA locus (can be focal or arm-level) or increased mRNA expression through other means.  The MYC oncogene is an example of an oncogene with characteristic activating mutations.

Tumor suppressors undergo inactivating mutations which can occur in a number of ways.  Truncating mutations (that alter the read cDNA read frame or generate a premature stop codon) are the most characteristic.  Other examples could include mutations in areas such as ligand binding domains that render the full length protein inactive, at least for a subset of its functions.  

It is difficult to compare the strength/dominance of an effect because the function of individual proteins is quite different.  That said, the "two-hit" hypothesis suggests that tumor suppressors do require inactivating mutations or deletions on both strands in order to be lost in the cell.  Genetic disorders that cause cancer, such as neurofibromatosis or Li-Fraumeni are the poster children for this theory.  Here the genetic background provides the first "hit" with full penetrance (100% of cells) and a second hit is all that is required to generate a tumor.

The reason two hits are (canonically) required on a tumor suppressor is because the remaining copy of the tumor suppressor gene is still functional.  For an oncogene, constitutive activation of one copy of the gene is all that is required to create an unregulated protein product.  I.e., because the oncogene is already near maximally active it does not need to be activated again on the other strand.  

This does not mean however that loss of a tumor suppressor on a single strand cannot lead to haplo-insufficiency that promotes tumorigenesis, or that a given mutation on one strand cannot create a dominant negative effect (this is known to occur in TP53 mutations).  So two hits are canonical to silence tumor suppressors, but not required.

I've linked an example of a dominant negative tumor suppressor mutation as well as a review on Dr. Knudson's two-hit hypothesis below.

http://www.nature.com/onc/journal/v23/n13/abs/1207396a.html

http://www.nature.com/nrc/journal/v1/n2/abs/nrc1101-157a.html

Thank you !