The Biologian: What Makes Something Sticky?

Bare bones answer: electrons being attracted to protons.

Molecules are collections of atoms. Within that molecule the distribution of protons (positively charged) and electrons (negatively charge) is not even. Often, but not always, there are more negative charges on one side and more positive on another. The negative side wants to stick to another positive sided molecule and vice versa. Broadly speaking these are called van der Waals forces or intermolecular forces if the attraction does not share electrons (well, for the most part...).

Chemists and physicists further break those broad categories into dipole-dipole forces (between molecules of differing charge distribution), hydrogen bonding (also a dipole molecule, but with a higher contrast between positive/negative), and ionic interactions (between ions), London dispersion forces (temporary redistribution of electrons resulting in charge asymmetry) and others shown below.

You can see these forces in action with tape. Stick some tape on a substance and then pull it up and some of whatever it was stuck to will invisibly now be on the tape in the form of stolen electrons (this shows there was at least some electron sharing, but not to the level of covalent or ionic bonds).

These two pieces were stuck together so that they now have the same charge and therefore repel each other

One piece was stuck to something and then removed taking some electrons with it. It's attracting to the tape that wasn't stuck to anything and because of that didn't pick up any extra electrons and therefore has more of a positive charge.