The difference between elementary and concerted reactions
The difference between SN1 and SN2 reaction mechanisms illustrates this quite well.

An SN1 type of reaction can be decomposed into two elementary steps:


formation of the carbocation
substitution of the nucleophile


The rate determining step is the formation of the carbocation, but we do have two distinguishable steps and a reaction intermediate - this is where the other mechanism comes in.

Now, when SN1 is not possible due to - for example - the carbon atom in question being primary, we can have a concerted mechanism - both of the steps mentioned before occur simultaneously, thus avoiding the energetically unstable primary carbocation. 

This process can not be broken down into single steps, as it is necessary that the formation and the substitution occur simultaneously. Also, there is no reaction intermediate - hence it is a elementary reaction, where two things happen in a 'concerted' way in a single step. 

Also, the experimental rate law will show that the rds is different and the rate depends on both reagents.