The lights are on
The bosses in God of War III are somewhat challenging even on God difficulty, and become insanely difficult on Titan and Chaos. Still, they are very much defeatable, and, like anything, can be countered with effective strategy. I'm writing the guide as if the reader is playing Chaos difficulty (the highest) but the general strategy can be used on any difficulty. Note: spoilers ahead, obviously, since I'm going to have to list the bosses.
1st Boss: Hippocamp
Otherwise known as the Water-Crab-Horse-Appendage thing. No, unfortunately you don't ruthlessly slaughter a camp of hippos. But I digress...
The first phase involves a straight-up fight with the Hippocamp, who has a single attack - a water spout. Nonetheless, this will either kill you or leave you very nearly dead on Chaos, so you must dodge every one of these. To accomplish this, stick to either the left or right side of the boss area, then roll to the opposite side whenever the Hippocamp begins to raise and shake it's head. Other than that, keep hacking.
The second phase occurs when Gaia flips her arm in pain. You are now hanging from the arm of Gaia, and the Hippocamp will only attack using either a left-side or right-side claw attack. He broadcasts which side he will attack by raising that side (or lowering, since you are upside-down at this point) for a brief moment. To avoid, simply move to the opposite side while hitting X (which speeds up Kratos). Getting hit will shave off about half of Kratos' health.
After sustaining enough damage, Gaia's arm catches onto a rock on Olympus, and Kratos is left climbing along it. The Hippocamp regains it's massively damaging water spout attack again, while retaining the left-and-right side claw attacks. Stay in the middle of Gaia's arm, hacking away until either the head rises and shakes (which signals a water spout) or either side or the Hippocamp rises up (that side will attack with it's claws). In the event of a claw attack, again, move to the other side. If a water spout attack occurs, simply move horizontally from the position that you were in when the attack animation began.
The fourth phase will occur after enough damage. It's essentially a repeat of the first phase, so repeat that strategy. After thoroughly ruining it's day, a O button command will appear over the Hippocamp's head. Pressing O will initiate a cutscene where you must press the buttons shown quickly enough, or suffer damage and redo the cutscene. Succeeding will take the Hippocamp's jaw off and end the boss fight!
2nd Boss: Hippocamp: Round Two
Far from dead, the Hippocamp is harder than before. In the first phase, the Hippocamp has it's trademark water spout attack, a close range clawing attack, and an attack where it drags itself forward before delivering an unavoidable water spout to Krato's face. For the water spout, just roll out the way (just like every other time). For the close-range claw attack, which is usually delivered mid-combo at a remarkably inconvenient time, pay attention to when the Hippocamp's claws raise sharply and roll directly backwards - this attack has less warning than any others thus far. Finally, when the Hippocamp drags itself forward, stay in the middle where neither of it's claws can hit you, and deal as much damage as possible - if enough damage is sustained, the boss will rear back in pain and not perform the water spout attack. If worst comes to worst, use a magic attack to both deal damage and become invulnerable against the water spout.
You'll get a O prompt, further wounding the Hippocamp. Finishing this context-sensitive cutscene will lead into phase two - which is identical to phase one, save that the boss gains another attack - when it's entire body rises upwards, run away! Afterwards, when it comes crashing down, it will create a shockwave which must be jumped over. Other than that, just hack'n'slash like you were before and a second prompt will appear, which causes Kratos to rip open the chest of the Hippocamp. After this, simply walk up to the ripped-off spike and lob it in the chest, and the Hippocamp is down for the count (or at least until...)
3rd Boss: Poseidon, God of the Sea
The fight against Poseidon starts on Gaia's head, with two Hippocamps (God, those things never die!) clinging to her. You'll need to attack both until each falls off, but Poseidon himself makes that difficult. He has two attacks, and multiple variants of both attacks - he summons lightning (looks like he likes to copy his brother) and attacks with even more Hippocamps which act as extra arms. Whenever the camera zooms to either side, he's going to try to strike you with his Hippocamp-arms. To avoid, simply roll to the other side. His other attack, a lightning strike, will occur shortly after half of Gaia's head is illuminated by a cyan static effect. Roll out of the cyan area to avoid the strikes. He can strike either the top, bottom, left, or right halves in this manner.
After knocking off both Hippocamps, you have to swing to Gaia's hand (R1) and initiate the second phase of the battle - the fight against Poseidon proper. He has four attacks you'll need to look out for, but only two are really a threat. First, he'll raise his fist and pummel the area directly in front of him - this does a lot of damage, but it's so slow that it poses next to no threat. Next, he'll slowly raise his trident and then violently thrust it, which will one-shot you if you don't back up. Again, this attack is very slow, but don't touch the trident - even after he's thrust it into the ground. His next attack involves a three-thrust combo with his trident - he'll attack the area directly in front of him, then the middleground, then reach around to attack the back. I suggest hanging around the back area for the first two hits, then moving to the front for the final attack. His final attack is an arm swipe which deals moderately high damage and is relatively fast. Roll back to avoid this, though I still find it difficult to avoid - it has a long range. Avoid these four attacks and wail on Poseidon in the intermissions between them. When a O prompt appears over his head, climb up the vines and complete the onscreen prompts to advance to phase three.
Phase three is essentially a repeat of phase one, but with harder-to-avoid lightning. Instead of striking one-half of the area, Poseidon now strikes either the left directly followed by the right (or vice versa) or the top third, middle third, and bottom third in sequential order. What this amounts to is that you'll have to roll away from the initial cyan static to avoid the first strike, then roll back to your starting position to avoid the second (and third, if applicable) strikes. Other than that, his Hippocamp attacks are unchanged. So wail on those claws until phase four begins...
Phase four simply involves swinging from claw to claw (R1) and finally hitting a final context-sensitive button at the end to free Gaia's fist. She'll Falcon Punch Poseidon, and Kratos is launched from her arm, going directly through the watery creation of himself which Poseidon made and grabbing the flesh-and-blood god that was within. After the cutscene, simply follow the button presses to watch a satisfyingly gruesome end to the annoying boss.
4th Boss: Hades, God of the Underworld
Hades is definitely the most resilient boss thus far (unless you count all three boss battles as part of the larger Poseidon fight). He has ten phases (more or less, depending on what you define as a 'phase') though fortunately the later ones are much easier. I'm tired, so I'll finish this guide later.