lifeaftr40 has offered good info.
Hydrocodone is one of the active ingredients in Vicodin, LorTab, Norco, etc. It also has tylenol mixed in.
Oxycodone is one of the active ingredients in Percocet, which also has tylenol. Oxycodone is the sole active ingredient in OxyIR, OxyContin, etc. It is the same oxycodone that is in Percocet.
Oxycodone is 50% stronger than hydrocodone.
Morphine is relatively equal in potency to hydrocodone, but is prescribed alone w/o tylenol mixed in. Many think of it as stronger than hydrocodone, but clinical studies show otherwise.
Above morphine and oxycodone are even more potent opioids.
For you, if you are in need of something stronger, I would recommend Norco, which is hydrocodone 10mg and tylenol. This is stronger than your Vicodin, but uses ingredients that have worked for you at some point at least, right?
If absolutely necessary, the next step up is probably Percocet with 5mg oxycodone and tylenol.
You want to be careful going up in potency with short acting opioids because tolerance develops very quickly, leaving you in need of ever stronger meds. If you have chronic pain then it is good to get on a long acting med. This helps keep your blood level of opioid stable and avoids the ups and downs of short acting meds.
What kind of pain do you have and how bad is it?