Recently I was emailed a simple question, "can mosquitoes make dogs sick?" The short answer is "yes". Thankfully there aren't too many diseases to worry about.
Of course the biggest concern is heartworm disease. Mosquitoes are the only vector of this disease and part of their larval development is in the mosquito. A single mosquito bite can infect a dog with heartworms and lead to a potentially fatal disease. Most vets talk a lot about this disorder, especially in the southeastern quarter of the US, so this is probably not a huge surprise. However, some people aren't aware of how dogs can get the disease so it's important to review it.
There is another disease that dogs and cats can get from moquitoes, but we're still trying to understand it. West Nile virus mostly affects humans, horses, and birds, sometimes with fatal effects. But there have been cases reported in pet dogs and cats, though thankfully it doesn't seem to cause as serious illness in these species and isn't as much of a concern.
Lastly, some cats can develop an extreme allergic reaction to a mosquito bite, resulting in severe inflammation and ulceration. Though it looks bad this is usually limited to the area of the bite and can be treated with antihistamines and antiinflammatories. It should still be seen by a vet but shouldn't be considered life-threatening.
For a little more information, the US Food and Drug Administration has some details on their website.