Great question! It turns out that the answers to both of your questions are connected. Oceanic crust and continental crust have different chemical compositions and as a result they have different properties, including density.

Density is defined as mass per volume - you can think of this as being how heavy an object is for a given size. For example, one cubic foot of steel is heavier than one cubic foot of water. If you were to drop a piece of steel into a glass of water, it will sink rather than float on top. However, styrofoam has a lower density than liquid water, so it would float on top of the water.

Oceanic crust is basaltic in composition. Basalt is a rock that forms when the underlying mantle partially melts, then crystalizes. The chemical composition of the rock formed from partially melting the mantle is different from the mantle itself because some elements prefer to partition into the liquid, whereas others stay in the solid. The density of basalt is around 3 grams per cubic centimeter. In contrast, the mantle is made of peridotite, which has a density of about 3.3 grams per cubic centimeter. Therefore, when the mantle partially melts and re-crystalizes as basalt, the oceanic crust will sit on top of the mantle.

Continental crust has an even lower density than oceanic crust because it is formed through multiple cycles of partial melting, leading to a granitic composition (its density is about 2.7 grams per cubic centimeter).

The asthenosphere is the portion of the mantle that flows, allowing the brittle lithosphere above it to move in a process called plate tectonics. Over time, oceanic crust cools, and therefore its density increases. When old oceanic crust collides with continental crust or with younger (and therefore less dense) oceanic crust, this pushes the old oceanic crust underneath the other crust due to its higher density. At this point the old oceanic crust is cool enough that its density is higher than the mantle it sits on top of, so once it starts to bend it continues to sink into the mantle in a process called subduction.

Scientists hypothesize that after being subducted, some oceanic crust sinks all the way to the bottom of the mantle. Eventually, nearly all oceanic crust is recycled in this process (a very small amount of oceanic crust is preserved on the continents - these deposits are called ophiolites).

Interesting question!
Oceanic crust is made out of a dense rock called basalt that is composed of the minerals plagioclase feldspar and pyroxene. Continental crust is made out of granite which is composed of the minerals quartz and plagioclase feldspar. The minerals that make up oceanic and continental crusts have different densities; oceanic crust is generally more dense than continental crust. Both oceanic and continental crusts are generally less dense than the asthenosphere, so most oceanic and continental crusts "float" on the asthenosphere.

Oceanic crust forms from hot magma rising to Earth's surface at divergent plate boundaries...over long periods of time, oceanic crust cools, and becomes more dense as it shrinks. Ocean water can be trapped in ocean crust, contributing to its density. Because ocean crust is more dense than continental crust, it will sink beneath continents if tectonic forces push the oceanic crust into the continent. This plate boundary where one type of crust sinks beneath another is called a subduction zone, and it is most commonly ocean crust sinking beneath continental crust or less-dense oceanic crust. In these subduction zones, the oceanic crust that is sinking beneath the continents is being pulled into the asthenosphere or deeper into Earth's interior in some cases. Ocean-continent subduction is happening today all around the Pacific Ocean, and is being actively studied by geologists off the coast of the Cascade Range (Washington, Oregon, and California), and off the coast of the Andes in Peru and Chile.

Continents float. This is because they are made of granite, which is less dense than the rocks of the asthenosphere.

Ocean crust does not float. However, the asthenosphere is very, very, viscous, like honey or tar, only much much moreso. This means that the skim of ocean crust slowly slides off of the top of the mantle and down, leaving a fissure where the break in the ocean crust is. This fissure is the mid-ocean ridges that produces new ocean crust.