Primary Specializations:

Unless otherwise stated most of these have ground and space utility and can be used as a secondary spec in lieu of one of the dedicated secondary specs down below (albeit instead of having all four tiers of perks active you only have the first two).

Intel: 

Usually selected as a primary spec for DPS because the tier 3 perk Space Flanking gives bonus (cat2) damage. It's passive boost is to defense which should decrease the amount of hits you receive.

Command: 

Good secondary for torpedo builds because it has a tier 1 series of perks that bring expose/exploit armour penetration de-buffs for torpedoes. Also good for tanking builds because the passive increases your hull capacity. Very hard to justify using this as a primary spec tho given the other available options.

Miracle Worker:

Great for tanking and healing builds for ground and space because it provides numerous perks for healing and damage reduction and its passive increases your hull re-gen in combat. Very niche but melee ground builds benefit from having this as a primary as well.

Pilot (Space only): 

Good for a different type of tanking that's focused on being nimble and evading damage entirely. Most of the perk points give some kind of defense boost so you'll increase your chances of not taking hits at all vs other tanking styles where the whole point is to take hits. Good for speedy, small escort ships. The passive bonus boosts turn rate by a significant amount, usually small escort ships don't need this passive boost but they benefit from it the most.

Temporal: 

Pretty much best in slot for exotic/science DPS builds. The passive boosts your EPG stat by a significant amount and the perks themselves revolve around getting the most out of your exotic damage abilities.

Secondary Specialization:

Strategist (Space only): 

Secondary only specialization that revolves around the use of Threatening Stance (a toggle ability everyone gets) and the various perk points depend on whether you have that ability active or not. This makes it a very good secondary specialization and is usually regarded as BIS for the current meta because it can be used for DPS builds or Healing/Tanking builds.

Commando (Ground only): 

Pretty much the best in slot specialization for ground content.

Constable: 

Hard to see a use for it in regular PvE content but the niche it holds is great buffs against single-target opponents. Unlike the other two secondary specs, Constable's perks are active in ground and space maps.

You can only switch specializations on social maps and not in combat. This means that a key limitation of the specialization system is that if you hyper-focus on say, space content, by picking space only primaries and secondaries, your ground game will be weaker as a result, and vice versa.