What do you call the ability of an atom in a chemical compound to attract electrons from another atom in the same compound?
Atoms in chemical compounds are bound together by Bonds, which are caused by the sharing of electrons between atoms. Every atom has two or more electrons orbiting the nucleus in their Electron Shells (only Hydrogen has a single electron). These electrons protect the nucleus and have an electrical charge, which can repel or attract with other electrons to form either new atoms or chemical compounds. Atoms can either join electrons with other atoms using Covalent Bonds, or remove electrons from other atoms, causing a positive/negative attraction between the two atoms: this is called an Ionic Bond.
An example of an Ionic Bonded compound is Sodium Chloride (NaCl) which is table salt; a Sodium Atom loses one electron to the Chloride Atom, which results in a positive charge for the Chloride and a negative charge for the Sodium. The two atoms are thus attracted to each other, but can still be split up by other atoms if they are added to the compound.
While a covalent bond is relevant here since it involves sharing electrons between two atoms, I suspect you are talking about electronegativity. Electronegativity is the measure of a particular element's general affinity to attract a negative charge (i.e. electrons) towards its nucleus. In general, electronegativity increases across a periodic row (left to right on the periodic table) and decreases down a group (top to bottom). This makes fluorine the most electronegative element on the periodic table.