I'll try my best to explain this, though, if I'm wrong, people please call me out.
As most people have mentioned, interest rates are the rates that people borrow money.
Lower rates = lower cost of borrowing Lower cost of borrowing = more borrowing more borrowing = more spending more spending = growing economy (one of the ways GDP is calculated is the expenditure approach which tallies up factors such as how much consumer spending there was in a given period. Higher levels of spending means larger GDP)
So that's the bare basics of it. To give a somewhat more detailed, but still broad, explanation of this, I'll try to give a brief explanation of how it affects equities, currency and market as a whole.
How it affect equities:
Generally speaking, lower interest rates drive up the price of equity valuations. This is part of the reason you see the run up in the Dow, S&P, etc. Why? Because of how equity is valued.
I won't go into discount rates and what they are too much, but basically, when interest rates are low, the yield on government securities like T-bills are low. The yield on 3-month T-bills is considered the risk-free rate. Risk-free rate is basically saying that you are investing with zero risk. This is because the US government has never defaulted. So when you invest in T-bills, your return is guaranteed. Essentially, any other investment outside of that carries risk.
So let's say you have this stock and you want to know how much it's worth. Well how do you do that?
One way (and this is an over simplification of this) is to look at how much t-bills are worth and what their yields are. Next you look at how much how much risk the stock carries in comparison to the t-bill. Well..as we said earlier, the T-bill is risk-free, so you have to be compensated for investing in something that's not so risk free like this stock you're looking at. One way to compensate you, is it price it so that it's an attractive investment. Essentially, this difference in risk, the risk-free rate, and the rate you expect the stock to return all go into valuing that stock. By the way the model is set up, lower interest rates lead to higher stock value. This is referred to the capital asset pricing model (CAPM).
the key takeaway: when interest rates are lower, the rate that gets plugged into the model to determine equity value makes it so that equity is worth more. Hence... stock prices go up. Of course there are other things that affect stock prices (some may have a larger weight on the overall price) such as future outlook, consumer confidence, etc.
Currency
How does it affect currency? Well, when you lower interest rates, what you're essentially doing is making it cheaper to borrow the dollar. When you make it cheaper to borrow the dollar, the currency naturally weakens. This is not necessarily a bad thing (despite Sarah Palin's comments on keeping "the dollar strong"). So higher interest rates make the value of the dollar go down; lower interest rates make it go up. Of course, other factors come into play as well.
Also, if you're curious on why a weak dollar is not necessarily a bad thing - when you have a weaker dollar, it improves things such as exports. When the USD is weak compared to, let's say, the Yen (Japanese currency), each Yen gets you more Dollars. This means, Japan can buy stuff from the US much more easily. This leads to Japan increasing their imports, and US increasing their exports. That's just one effect.
Market as a whole
Not much to say, but that you keep interest rates low in the hopes of stimulating the economy. You see the need for more consumption, higher levels of exports, etc. to grow the economy and the way to do it is to lower the interest rates. It's a fine line, because if it persists for too long, you end up having hyperinflation, where stock prices are going crazy, labor rates are going up, price of everything is going up, currency crashes, etc etc. So that simple fact that interest rates make it easier to borrow money has far-reaching effects. Not just on the consumer looking to get a loan.