Androids are very tough. They are the best at the physical stats and completely immune to psionics. Although they will eventually be surpassed by human soldiers that do manage to outlive them, their immunity to psionics and Brainsuckers is not to be underestimated. Use them at every opportunity.
Androids make excellent shock troopers and are ideal for sending into UFOs first - not due to their expendability, but due to their immunity to brainsucking and psionics. Androids are also highly capable in combat and often start off with superior physical statistics, making them excellent heavy weapons troopers early on - they can easily lug around an autocannon or other heavy weapon, plenty of spare ammo, and a handful of grenades without any noticeable speed reduction... and then run around the map for hours. Not only that, but their starting morale is usually 80 or higher, making them unshakeable in combat except in the face of vastly superior numbers and weaponry.
They are highly dependable and very durable (and some with a more macabre outlook on life would say, expendable). Androids can go so far as becoming body shields to protect their other training comrades, as they need not be in top form at the end of the day seeing as they are incapable of using training facilities. As androids are mechanical in nature, they are mostly ignored in the field of psionics, which means that though they cannot use it, they are also immune to it. That maxed out Psi Defense stat is just for show, as psi-capable enemies ignore them completely.
Not only are they immune to brainsuckers (though they still cause brainsucker pods to hatch) but brainsuckers will actually not regard them as legitimate targets! The brainsuckers will attempt to flee from Androids or stand there looking extremely confused if there are no other human/hybrid agents in range. It looks mostly human and it sounds mostly human, but it's not a hybrid... something isn't right... the brainsucker's tiny mind cannot cope with this!
Contrary to popular belief, Androids reportedly do gain experience and improve their statistics, but at an incredibly slow rate - androids may not experience any increase until well towards the end of a campaign even if they have been with it from the very start.
Creating all-android squads or even using them exclusively for your ground teams has it's advantages and disadvantages, but usually a mixed force of humans, hybrids and androids is more versatile and effective. Androids can chase down enemies much further and faster than humans (and much much further than hybrids) meaning a mixed squad may end up being left behind by their android squadmates. Additionally, they are only surpassed by human soldiers who survive a very long time, which is generally a rare occurance among human soldiers.
Androids seem to take only 20% normal damage from Stun weapons, and no damage from Stun Gas (as they do not breathe), as well as only 60% damage from Incendiary sources. (data from Apoc'd)
Maintaining good relations with S.E.L.F. is key to ensuring a steady supply of potential android recruits.