Increases at ISO allow you to take pictures with lower light, yes.
But this comes at the expense of image quality.
Below is a 100% sized crop of my Canon PowerShot A75 at the default ISO 50:
And here'* ISO 400:
That is known as "noise", or "grain" if you still shoot film. Digital SLR cameras should be able to shoot comfortably up to ISO 800, and at 1600 and 3200 they begin to show some noise, but much much less at 3200 than my A75 at 400.
Each time the ISO number doubles, you need half the amount of light to capture the same image: or "one stop" in the exposure scale. For future reference, if you can use a tripod and a low ISO setting, do that.
Generally, as a rule of thumb, you should be using a tripod (or find a stable surface) when shutter speeds get below 1/60 of a second. Does your camera tell you the shutter speed and aperture it uses when you go to focus the image?
asterisk: Sometimes, grainier images are desired for added effect - primarily when shooting with black and white.