A brownish hue (this one is easy)
Take a good look at the color of your wine when your pour it in a clear glass. If your red wine has a brownish hue, it's past its prime, and a deep yellow or brownish straw hue in your white means it's probably oxidized. How about if you get some fizz in the glass, but it's not your favorite bottle of sparkling wine? It's had a second fermentation after the bottling, and this bottle should be dumped.
The simple smell test
Give your wine a good swirl and take a good long sniff. Musty, moldy, vinegar, sauerkraut or rotten egg smells are all signs the wine has gone south. Dump it.
Sip that wine, finally! Do you like it? If so, great! Try to point out the things you love about it, like the aromatics you're able to identify. If you hate it, remember why. A vinegar taste similar to that awful aroma you may have picked up is a sign to dump it. Picking up a really, really sour note on your sip, or just no more luscious fruit notes at all (aka flat wine)? You probably left that wine open waaaayyy too long...why did you have leftover wine anyway?!
Bad spot for storage
A wine bottle that's warm to the touch probably wasn't stored properly. Invest in a wine fridge -- it's worth it! Or store your wine in a dry, cool place like your basement. The kitchen is the worst place ever because of the varying temperatures. Don't do it!