A category of turian soldier, loosly translating to "execution squads".
The hastatim are a specialized brigade or division - or even an entire corps, depending on the size of the Legion - that has been trained to deal with insurgency/guerilla warfare, close-quarter-combat, take-and-hold operations in an urban setting, guard operations, and general removal of troublemakers.
The reasoning behind the hastatim is that of the general Hierarchy war doctrine; you get a warning to surrender or leave an area, if you do not comply you're regarded as an enemy. As all turians over 15 years old are considered military combatants, all enemies are assumed to be armed and dangerous and lethal force is used to deal with them. By contrast, surrender is met with recognition that one has placed their life in the hastatims' hands and they will protect it to the best of our abilities, even risking their own lives.
The role of the hastatim was first conceived by Primarch Eremus, as a way to quell growing rebellious behavior after he introduced some major economic reforms. Originally the hastatim were part of the Police; a more aggressive, proactive special intervention force. From there they grew; now every major Legion has at least one brigade, division or even a corps of hastatim (generally speaking if it's just one brigade it will be incorporated in a general infantry division.) Smaller legions sometimes have smaller Hastatim units; for example the legions of Gothis only have Hastatim battalions.
Other particulars of the Hastatim include the fact that they're generally trained harder than GI units and are considered 'harder' and more 'ruthless'. This is mostly due to the fact that they're deployed in domestic operations (against separatists), so they have a certain image as the group that kills other turians in Hierarchy space.