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Electronegativity is an atom's 'desire' to grab another atom's electrons.

Period - electronegativity increases as you go from left to right across a period.

Why? Elements on the left of the period table have 1 -2 valence electrons and would rather give those few valence electrons away (to achieve the octet in a lower energy level) than grab another atom's electrons. As a result, they have low electronegativity. Elements on the right side of the period table only need a few electrons to complete the octet, so they have strong desire to grab another atom's electrons.

Group - electronegativity decreases as you go down a group.

Why? Elements near the top of the period table have few electrons to begin with; every electron is a big deal. They have a stronger desire to acquire more electrons. Elements near the bottom of the chart have so many electrons that loosing or acquiring an electron is not as big a deal. This is due to the shielding affect where electrons in lower energy levels shield the positive charge of the nucleus from outer electrons resulting in those outer electrons not being as tightly bound to the atom.