Could be a number of possibilities...
Clear your browser cache. Typically if something updates correctly in one browser, and not in another, clearing the cache will fix it. Basically the browser may be loading from memory, instead of loading from the actual page. Usually Chrome is horrible at this, but it can happen with any browser. A quick search for "Clearing __your browser here's__ cache" will get you the directions for your particular version.
Sometimes a small error in your html can cause a display problem in one browser and not another. Basically all browsers are programmed to try to display a page, errors or not, some (OK, pretty much all) are just much better than IE at guessing what was meant when they run into code issues. Run the validator at: Ready to check - Nu Html Checker to get a listing of any errors. Cleaning them out could fix it.
Bad coding methods, like using absolute positioning and overlapping a linked image with the empty space of another absolutely positioned element can "steal the click" from your linked image. Basically causing the browser to think you're clicking on an empty container, rather than the link below it. If you're using APDivs/Absolute Positioning, this one is probably the hardest to fix because it requires a rethinking of how a page is designed.
There are other reasons, but without seeing your actual page in our browsers, it's hard to say what else it might be.