It's a simple question.. What do continents "lay" on? Do they float on water? or are they huge bodies that "emerge" from the sea floor/bed? are they connected to the bottom of the oceans? Hope the question is clear. Don't be afraid to be thorough and scientific in your answer, I'll appreciate it and do everything to understand it.
Matan, the continents where we all live "float" on the Earth's mantle. The continents are made out of relatively brittle rock called the "Crust" and the mantle is made out of much more ductile material. The mantle, however, is NOT liquid. It is just much more ductile than the crust so, in geologic time, it can flow (like silly putty). Also, the mantle is much more dense, so the crust doesn't sink into it by gravitational/bouyancy forces alone.
Think of it as a creme brulee. We live on the little hard crust on the top. And there just happens to be a little bit of water on that crust that covers the less-elevated parts.
Now, a little more technical, the crust that makes up the continents and the crust that is under the deep oceans are actually compositionally different. This is because the magma that solidifies to form each of these crusts travels, by partially melting its way up, through different materials and different thicknesses. The consequence of this is that the crust making up the continents is less dense than that making up the bottom of the oceans.
The image above shows the process by which we get new (left) oceanic crust, (right) continental crust, and (middle) a special type of crust like that in Hawaii. Note that when two tectonic plates collide, the more dense one will tend to subduct under the other.
[This answer has been highly critisized for using only the oversimplified descriptions used in school textbooks as an answer; the original answer is included only to make sense of the comments]
(I'm not a professional geologist; this is from my personal studies) Lava, essentially. The Earth is molten when you get farther down, and the surface of the planet is made up of large plates of stone, which very slowly get pushed around by huge swirls of lava beneath them.
Note that this is a very simplified explanation, but it seems to answer your question.