How the periodic table is organized
Elements are arranged in order of increasing atomic number. Elements in the same group (column) have the same number of electrons in their outer shell, giving them similar chemical properties. Elements in the same period (row) have the same number of electron shells, but the number of outer electrons increases across the period.
Trends across a period
Moving left to right across a period, atomic radius decreases (more protons pull the same number of shells in tighter), ionization energy increases (electrons are held more tightly), and electronegativity increases (atoms attract bonding electrons more strongly).
Trends down a group
Moving down a group, atomic radius increases (an extra electron shell is added each time), and ionization energy decreases (outer electrons are further from the nucleus and more shielded, so easier to remove). This is why reactivity of Group 1 metals (like sodium and potassium) increases going down the group, while reactivity of Group 7 non-metals (like fluorine and chlorine) decreases going down the group.
Example 1: Explain why potassium is more reactive than sodium, even though both are Group 1 metals.
Potassium's outer electron is in a shell further from the nucleus than sodium's, and is shielded by more inner electron shells. This makes potassium's outer electron easier to lose, so potassium reacts more vigorously (e.g. with water) than sodium.