Ionization energy is the amount of energy needed to ionize an atom or kick an electron off. Different elements have different ionization energies, which can be calculated using approximations based on quantum mechanics or measured experimentally. Both of these are fairly challenging tasks. Fortunately, however, the periodic table exhibits some simple ionization energy trends. So if you have a group of elements and you want to figure out which has the highest ionization energy, you can often just look at the periodic table to find out.
Locate each of the elements you want to compare on the periodic table. If you have fluorine, sulfur and lithium, for example, fluorine is in column 17, sulfur is in column 16 and lithium is in column 1.
Determine which elements lie highest and farthest to the right. In general, as you go across the periodic table to the right, ionization energies increase. As you go up the periodic table, ionization energies also increase. To continue the example, fluorine lies highest and farthest to the right, so it has the highest ionization energy.
Rank the other elements in the group from highest and farthest to the right down to lowest and farthest to the left. This is the order of their ionization energies. To continue the example, fluorine has a higher ionization energy than sulfur, which has a higher ionization energy than lithium.
Notice that hydrogen is something of an anomaly -- its ionization energy lies between that of carbon and nitrogen, but it's usually placed in column 1. If hydrogen is in the group, treat it as though it were between carbon and nitrogen.
These trends, while useful, are broad generalizations with many exceptions. If you need to know the actual ionization energy of an element, it's best to simply look it up -- chemists and physicists have measured and calculated ionization energies for nearly all the elements. The data are widely available.