Historically, having a strong dollar has been good for the average citizen. After all, a strong dollar will go farther in an international market than a weak dollar, enabling the tourist ($1 = X international currency! Everything is so cheap here!) as well as allowing one to purchase international stocks for relatively little. However, recently the opposite has been true. Since late summer, a strong dollar has meant a dip in the stock market and visa versa. While at first this seems counter intuitive it does make logical sense for a couple of reasons.
First, just as the tourist is excited to buy foreign goods when the dollar is strong, foreign markets are eager to get the best prices they can on American goods. This means that when the dollar is weak our exports have a competitive edge. This confidence is reflected in the stock market when stocks rise. The best supporting example for this phenomenon is China. Economists have accused China of intentionally devaluing its currency to help their export driven economy (something the United States was also accused of recently at the G20).
Second, when the future is uncertain and the stock market takes a hit (as in the Irish bailout and the fear of the contagion spreading to Spain and Portugal) the dollar has tended to be at its strongest. This is because, like gold or oil, the dollar is seen as an investment for hard times. Similar to people stuffing their mattresses with greenbacks during the depression, investing in the dollar is seen as safer than the market at the moment.
However, the dollar and the stock market have only rarely been so closely tied before. It seems likely, therefore, that all of these explanations are only really applicable in a bear market. All we can do is speculate and wait to see if this fascinating trend continues.