The Carbon-Hydrogen bond is the covalent bond between a carbon atom and a hydrogen atom. Almost all organic substances have this type of bond, and they make complex chemicals such as polymers, and thus life as we know it, possible. And, somewhat ironically, the carbon-hydrogen bond does so much by doing so little.
The basics of covalent bonds are probably explained elsewhere either here or elsewhere with much more detail and math than I can muster. With apologies for anthropomorphizing, each atom wants to fill a certain amount of electron shells. Hydrogen has one electron, and needs to have two. Carbon has six, and needs four more. So what atoms do is share electrons, which allows them to fill their electron shells while keeping a net neutral electrical charge. For carbon to fill in its shells, it bonds with four hydrogen atoms, forming methane, which gives all atoms involved full shells, with a net neutral charge. Of course, all chemistry isn't this simple, and sometimes carbon is bonded to other atoms, and sometimes to other carbon atoms, but when it isn't bonding to another atom, it can pick up a small hydrogen atom to fill in the gaps. And since hydrogen can only bond to one atom, this bond also acts as a terminator on chains of carbon bonds.
In simple terms, atoms want to fill their shells with electrons, but some want to fill them more than others. For example, oxygen holds on to its electrons tighter than its partners do. To quantify this, Linus Pauling invented something called the electronegativity scale. Each atom is given a number from 4.0 to 0.7 that reflects its affinity for electrons, from the very hungry Fluorine to Cesium, which wants to get rid of its electrons. By subtracting the electronegativity between two elements, you can tell how even their bond will be. Since carbon has a number of 2.6, and hydrogen 2.2, the charge is for all practical purposes, even between them. This leads to two properties. First, because the bond is shared equally, carbon-hydrogen bonds don't show polarity, and aren't attracted to other molecules. That is why methane, with an atomic weight about equal to water, is gaseous at a temperature where water is solid. Also, the fact that a carbon chain has very little reactivity between its elements makes it very stable. Carbon needs to fill its bonds, and it could do that, theoretically, by forming a carbon chain with each carbon double bonded to an oxygen atom. However, such system would be unstable because the oxygen atoms would react with the carbon chain, breaking it apart. So only through filling up their extra bonds with the unobtrusive hydrogen atom can carbon form the long chains and polymers that are both internally and externally stable, thus allowing other groups to attach to the molecules. It is the somewhat boring nature of the bond that allows other, more complex, diverse molecules to form.
Because it is so omnipresent, the carbon-hydrogen bond is often not even written out in schematic diagrams of chemicals, they are usually left out, the viewer understanding that carbons that are not otherwise bonded have their remaining bonds formed with hydrogen.