What's in a name? Quite a bit. Even the type of name can tell you something because different types of names are associated with different types of compounds.
You don't need to be familiar with every element on the periodic table. However, you should be able to recognize the names of all the non-metals, plus a good many of the metals. You should be able to quickly scan the periodic table to find an element, so that you know when someone says "magnesium" that you are dealing with the element Mg.
Sometimes elements exist as diatomic molecules, such as O2 and H2. You should be familiar with the seven elements this is true for: hydrogen, nitrogen, oxygen, fluorine, chlorine, bromine, and iodine. So you should know that if someone says "elemental iodine" they are talking about I2.
Prefix names are used with covalent compounds. The name tells which elements are present. The most electronegative element is last in the name and has an "-ide" ending. The prefixes tell how many of each atom are in the formula. The prefixes used are "mon-" or "mono-" for one, "di-" for two, "tri- " for three, "tetra-" for four, "penta-" for five, "hexa-" for six, "hepta-" for seven, "octa-" for eight, "nona-" for nine, and "deca-" for ten. The absence of a prefix generally means there is only one of those atoms in the formula if it is a prefix name. The "a-" and "o-" endings of the prefixes are generally dropped when attached to "oxide."
Some examples of prefix names are dinitrogen tetroxide and phosphorus trichloride. The interpretation of prefix names to get a formula is quite straightforward. Simply write down the elements present and use the prefixes to show how many of each atom you have. So dinitrogen tetroxide would be N2O4 and phosphorus trichloride would be PCl3.
If a name contains more than one element and has no prefixes, it is the name of an ionic compound. Because the names sometimes do not give you any clues as to how many atoms there are, you have to figure that things out based on what you know about the elements, usually from their position on the periodic table. For example, when someone says "barium chloride" you should immediately identify this as an ionic compound, look at the table and find that barium always forms +2 ions and chloride is -1, and determine this compound's formula is BaCl2.
Sometimes names without prefixes will include a roman numeral. This roman numeral always represents the charge on the element whose name comes before it. So nickel (II) bromide contains nickel with a +2 charge, and bromide is -1. This makes the formula NiBr2.
These names may also include polyatomic ions, which you should know eight of. So, if someone tells you they have iron (III) hydroxide, you should be aware they are referring to Fe(OH)3.