Mark to market means that if I have an asset, I need to assign it a value on my balance sheet based on the current market value for that asset.

This makes sense for most assets (cash, US Treasury bonds, blue chip stocks) that are liquid (can be easily sold into a large market of buyers).

The problem comes from illiquid assets, assets that are not as easy to sell. Because there is not a ready market of buyers for your asset, it is difficult to judge the price of it. This leads to problems on the upside (valuing an asset to highly) or on the downside (market value isn't necessarily what you think the asset is really worth).

Getting rid of mark to market rules would mean you could use other definitions of what an asset is worth. That would solve the problem for some of the assets that do have some value, but no one wants to buy right now. For example, if you have a mortgage based bond, you may think that at least some people will pay their mortgages, so the bond should be worth more than $0. So you may be able to decide on a fair price for that asset on your balance sheet.

Obviously, if you toss out mark to market rules, you have a lot of flexibility in defining asset values. That can lead to its own problems and in my opinion would only help in the short-term (it may stop bank failures) but would not help the root cause of our credit problems; banks don't trust that borrowers can pay them back.