As I see it, equity is the difference in value of a property and any debt related to that property. The property could be real estate, a manufacturing facility, a stock/bond/commodity, anything of value.
Equity is one type of asset. Other assets are fixed assets (equipment), cash, accounts receivable, even good will. An asset is usually something that has a cash value of some kind. In other words, you could convert it to cash at some point by having someone pay you for it.
I am not an accountant, but I have worked with many financial systems over the years as a contract programmer. I think these are reasonable definitions in laymen's terms. An accountant or financial expert could give you a much more precise definition. I imagine that you're looking for a simpler explanation, though, since you could look both terms up in any dictionary.
An asset is anything you own. A girl's figure is one of her assets. you might own a house as opposed to renting one. There is no real equity in a girl's figure but in regards to the house, you might have bought a $100,000 house this morning with 100% financing from the bank. You own the house but owe the bank for the entire price leaving no equity for you. Once you pay the bank off, you have 100% equity. Now say you bought that same house at auction for $60,000 and it appraises for $100,000. You have instant equity of $40,000. Equity is how much of the total value of the asset that you own. It can be expressed as a dollar figure or a percent. You might also say, "I have equity in...a cotton farm (and not state the %) This may be because you are part of a real estate group and the group owns the farm. So you own some vague % of the farm. Aslo you gain no ownership or equity when renting.
I hope that helps.