I assume you have an understanding of what resistance is.
Let's say you have some wire. It has a certain electrical resistance (measured in $\Omega$). If you have an identical wire that is twice as long, it will have twice the resistance. If you have an identical wire that has twice the cross section it will have half the resistance of the original wire.
Now, physicists are usually not interested in a particular wire, but in the general case. If you are going to make a theory for current through a metal wire, you are not going to make that for say a $1~m\times 1~mm^2$ wire, because that theory would be somewhat narrow, limited to one specific case. (What if my wire is 1.5 m long?)
In order to get a more general theory you get rid of the length and cross section dependence, simply by studying the resistivity, $\rho$, which is defined through the equation
By this procedure you get rid of the size dependence and $\rho$ will depend basically on the material only (and temperature, etc). Also it explains the unit $\Omega m$.
A procedure like this is very common in physics and usually denoted by the word specific, i.e. you could call $\rho$ the specific (electrical) resistance.
Another example from physics is the "specific heat capacity", which is the heat capacity per kg. Just like the resistivity, the specific heat capacity depends on the material only unlike the heat capacity which is proportional to the mass.
Or an example from everyday life: If you go to a shop to buy 0.5 kg of apples you pay a price for them. If your friend goes to the same shop to buy 2 kg of apples he will pay a different price (four times as much as you), simply because he bought a different amount of apples. In order to avoid having to display prices for 0.5 kg, 2 kg, and all kind of other amounts, the shop just displays the price per kilogram (in physics language, the "specific price" of apples). So, while there is some interest in the price of 2 kg/0.5 kg of apples in certain situation (namely for you and your friend), for the general case, the "specific price" is much more convenient.
To answer your question, whether to use resistance or resistivity is basically just a question of context. If you are interested in a particular piece of resistor, you are likely concerned with its resistance. If you are studying more general things, resistivity is your quantity of choice.