Group
A group is a vertical column of the periodic table; elements in a group share the same number of valence electrons and similar chemical behavior.
Definition
A group is a column going up and down the periodic table, and elements in the same group act alike; there are $18$ groups. Also called a family, a group is numbered $1$ through $18$, and its elements have the same number of valence electrons, so they form ions with the same charge and react in similar ways; well-known groups include the alkali metals (group $1$), alkaline earth metals (group $2$), halogens (group $17$), and noble gases (group $18$). The shared valence configuration ($ns^1$ for group $1$, $ns^2 np^5$ for group $17$, and so on) produces similar chemistry, oxidation states, and compound formulas. Descending a group adds a principal shell, so atomic radius increases while ionization energy and electronegativity decrease, and groups $3$ through $12$ are the transition metals, whose partially filled $d$ subshells give them variable oxidation states and colored compounds.
Example
Helium, neon, and argon are all in the last column, group $18$, and none of them like to react with anything, which is why they are used in balloons and glowing signs. Every group $2$ element (beryllium, magnesium, calcium, strontium, barium) has $2$ valence electrons and forms a $2+$ ion, and magnesium and calcium both react with hydrochloric acid to release hydrogen gas. Group $17$ electronegativities fall from fluorine ($3.98$) to iodine ($2.66$) on the Pauling scale, so fluorine displaces chloride from solution ($F_2 + 2Cl^- \rightarrow 2F^- + Cl_2$) but iodine cannot, and all four form hydrogen halides $HX$ and sodium salts $NaX$.
Key Insight
If you know how one element in a group behaves, you can make a good guess about the others in that column. Reactivity changes down a group in a predictable direction: alkali metals get more reactive going down (potassium is more violent than sodium), while halogens get less reactive going down (fluorine is the most aggressive). The modern $1$ through $18$ numbering replaced the older IA through VIIIA scheme, but the Roman numerals survive because they equal the valence electron count for main-group elements, and either way the column is the fastest predictor of chemistry on the table.