I have a Dell Inspiron 620, Intel Core i5 CPU @ 2.90GHZ, 8GB RAM, 64-bit Win 7, and i have the basic Intel HD Graphics and need a different one because i do a lot of gaming.I dont have a budget either. also im running a 24" 1920x1080 resolution.
There are a wide variety of cards that you could use , however you are handicapped by the power supply which I believe is 300w. You should check to make sure.
Since you don't have a budget then you can start with getting a bigger power supply. Is your case a Slimline or Minitower. If you have a mini tower and you can get a 500w psu that will fit then you can get either a GTX 560Ti or an AMD 6850. A good thing for you to do is to contact Dell support and find out what you can upgrade the psu to. That will let you know if you can go for the cards that I mentioned or if you will have to get something lower. There are a lot of cards that will be better than the intergrated video you now have.