Cant Sign In To Secured Sites

I purchased a Dell computer last year. My kids were going on the computer and I started to get pop ups and it was running slow. I had my computer reformatted (I believe thats what you call it). When I hooked my computer back up, it ran great, no more pop ups and it was running faster than ever. One problem, I cannot sign in to certain sites. I cannot sign in to Ebay, Verizon, etc. I brought my computer back to my friend with the problem and he hooked it up and I was able to sign on everything there. My friend has adelphia and I have verizon. I called verizon and was on the phone with them for hours. Cked all settings, they searched and searched and no luck. My computer is only a year old and I have not been on because Im having this problem. Any suggestions what this could be. I have no idea why I would not be able to access any of these sites when I can at my friends house with powerlink. I want to mention that the disc for the settings that came with my dell computer was missing, my friend went on the dell sight and downloaded this, he says thats not the problem. Im stuck and running out of ideas what this could be.