My husband has ran a snake down the drain and the doen't seem to be anything stopping it. But as soon as we run water into the drain it begins to back up. It drains very slowly and if I run the dishwasher when it rinses it backs up into the sink. What can we do?

You drain my simply be blocked further down the line than he is reaching. Try a longer snake or remove the P trap and snake from under the sink. You could try a home remedy but it takes all night. Mix half cup salt, half cup baking soda and one cup of vinegar. Pour it down the drain and allow to sit over night. Tomorrow morning fill a bucket full of the hotttest water you can get, near boiling. Dump down the drain, 9 out of 10 times your problem will be greatly improved.

If Bals treatment doesn't get it then you got to get down and dirty.
Most likely your clog isn't in the trap but in the lateral line in the wall. Remove the "J" bend from the trap and send a snake up into the wall.
The good news is that you shouldn't have to send out over 6 or 7 feet until you run into the sanitary tee at the kitchen vent. Flush the line out with a sink full of hot water after you reassemble everything. Good luck, Tom