There are two ways of going about this achievement though, unless you are very proficient in combat in the Assassin's Creed series, this can be a little difficult to earn.
The first way of doing this is to select Virtual Training in the Animus' menu, choose the combat section and load up any of the "Kill Streak" missions. These will put you into an arena with an endless stream of enemies from which to earn this.
The other way of earning this, and probably the more beneficial one if you are focusing on the singleplayer game more than anything else, is to load up the "Crepi Il Lupo" memory from Sequence 2. If you mess up during this memory though you have to suffer through a 20 second cutscene every time you restart it and trying to skip it just takes longer than sitting through it.
Either way you approach this the combat is the same. Assassin's Creed Brotherhood, unlike the previous two games, emphasizes offensive rather than defensive combat, hence the execution ability. The first thing you need to do is kill your first target. Immediately upon his death move the left stick in the direction of the nearest enemy (or the most convenient target) and press X. Doing so, if they are in range, will initiate the execution move. This needs to be done an additional nine times in order to unlock the achievement. The thing to note is that, unlike the previous titles in the series, Assassin's Creed Brotherhood gives you the ability to cancel your animations in mid-attack in order to perform a counter move. Should an enemy attack you in the middle of an execution, which they will often do, simply hold down the right trigger and press X while they are in the middle of their attack animation to counter kill them. This might be a little easier to accomplish in the "Crepi Il Lupo" memory rather than in a Virtual Training session because there seems to be only one attack animation for them versus the various enemy types in the training sessions. The enemy attack animations in "Crepi Il Lupo" are quite drawn out (you have almost a second to react should they attack) so, if you can suffer waiting through a cutscene, you are good to go.
Jack and Geoff from Achievement Hunter earned the achievement on "Crepi Il Lupo" in the video below should you need a visual reference. Good luck!