Recently our lights have started to dim or brighten at random times. It sometimes, but not always, happens when the microwave is running. Other times it happens when nothing extra is being ran. Also I have noticed that it sometimes seems to happen when it is rather windy outside, but again, not always. It started happening about a month or two ago but now it seems to be happening more often.

Start by having your power company come out and confirm that they are maintaining a good sound connection to you

You have a problem with your service neutral. Classic symptoms.
The wind clue points to a problem at the transformer or the connections at your house.

Call your power company right away!

That's what I was trying to say but Stanforty says it better

Thanks. I have called the electric company and they are on their way to check it out. I hope it's something on their end! I'm a little more concerned now though because when my furnace came on the lights totally dimmed and almost went out when the furnace turned off it continued to make a humming noise for another minute and then when it stopped the lights brightened again. I'm totally clueless... thinking maybe the funace continued to make the noise because it was still trying to run but there was not enough power getting to it to keep it going? Guess I will wait and see what the power company says and if it's not something on their end I will have to call an electrician tomorrow:( Hoping for the best!

What was the outcome of the power companies visit? This does need to be corrected ASAP, as the dimming can damage motors such as refrigerators, and the brightening can damage anything by the abnormally high voltage.

They said that there was the proper amount of amps, voltage, or whatever it is coming into the house but they would have someone come out Monday or Tuesday and replace the service. That's nice and all but it's getting really bad with the dimming. I turned the furnace completely off because I'm afraid that something will happen that will cause a fire or something since it dimmed so bad when it turned on. I can't go without heat until Monday or Tuesday, it's in the teens here with windchills in the single digits. Next thing you know, my pipes will be frozen too. I'm thinking I should just get an electrician to come out but I can only imagine how much that's going to cost me on Super Bowl Sunday. But who's to say that when the electric company does replace the service that it's going to fix the problem. They didn't seem too confident that it would nor did they seem like I should be so worried about the issue.