The weight of the toilet has gotten rid of the ring that was there. YES you need a ring as it stops sewer gas from entering your home -- which can explode. Most new wax rings come with a cone attached to the ring. The cone helps direct the waste water down the drain.
After a new ring is in place your toilet should not leak. If it does then have the flooring under it checked for rotting wood etc, broken toilet hold-down flange etc.

Check the toilet exterior starting up as high as possible with your dry hand. Work down from there. The water could be coming from higher up.
A bad wax seal will leak only at the moment the toilet is flushed.
Do you have children who may have "bad aim"?

Thanks Toolbelt and bus_driver. There is no evidence of there ever being a wax ring, but my husband did say there were some sort of foam pads. We never have smelled anything unpleasant or noxious, so the pads must have been serving the same purpose as a wax ring.

No one with poor aim here, it's definitely water. In fact, the cat was drinking from the puddle on the floor ;)

There is definitely a leak from the front of the base, even without flushing. There is also a leak in the back that seems to be dripping from somewhere, probably the screws that attach the tank to the stool. Husband is going to try to fix it when he gets home tonight, but I thought I could maybe take a crack at it since I am at home today.

Thanks again!

kimberly-
It's hard to understand how you didn't have major leaking before if there was no wax ring. And, whatever those pieces of foam were, that is nothing that is ever used or should be used in a proper installation. You did the right thing by putting on a new wax ring. However, there are two things you need to be sure about:
1. The flange in the floor should be anywhere from flush with theexisting floor to about 1/4" above it. If it was below the surface of the floor, one wax ring may be insufficient to make the seal, and a second ring should have been used on top of the first one. When the toilet was set on the ring, you will feel resistance and the toilet will sit a little off the floor until the wax spreads out and allows the toilet to settle to the floor. If you did not get this resistance, that's another sign that the wax wing was inadequate.
2. If the toilet was not positioned properly when resetting the toilet -- exactly centered as the toilet was lowered, and level -- the new wax ring could have been damaged.

Also, if you have water coming from the point where the tank sits on the rear of the bowl, it's possible that the bolts are not tight enough. When you try to rock the tank, does it seem pretty firm, or does it wiggle some?

Please report back when you have investigated further.

Two trivial things:

Here's an easy check for us lazy people: first dry the floor very well and then put a ton of colored food dye in the tank that is fully filled and has no more water running. That will tell you if the tank is leaking externally. Hopefully the flapper is good and tight so the bowl water stays clear. Once you flush, this test is over!

Also it's really easy to smush a wax ring (not the only sealing method, BTW) when horsing a heavy toilet in a tight space onto two obscure bolts you can't see. Use two people.

Oh, one more. We can thank HD for this one. If there was a "bungled" repair to the waste line, and the floor flange was replaced with an insertable non-glue (the sort with a rubber slip seal) new floor flange onto the inside diameter of a 3 inch waste line, that handy dandy little plastic funnel/sleeve/cone may not fit into that tight little waste hole. Hard to see but it might show up right away as a rocking toilet.

that dye trick is killer. used it a couple times with success.

Posted by kudzu9

kimberly-
It's hard to understand how you didn't have major leaking before if there was no wax ring.

Fenco, Sani Seal are two that I can think of off of the top of my head that are not Wax Seals and have been on the market for years.

Kim mentioned foam. That would be a Sani Seal brand toilet seal.

hippy-
Yes, I know about waxless seals. However, the OP's first post seemed to indicate no seal; the second referred to "foam pads" [plural], which sounded like some hack job plumbing, rather than a SaniSeal to me...but maybe you're right.

kudzu9

I understood the "foam pads" [plural] thing and off of the top of my head, Sani Seals are stackable.