How to get and summon Titan
First thing's first: as with all summons, you'll gain access to Titan as part of the main story. He's the first of the astrals to come to your aid, and you'll get him to do so via a big battle in Chapter 4 of the game.

Titan helps you out in wide open areas. This means he isn't going to show up in fields and the like, but he will show up out in the overworld. He has two potential moves: he'll either punch the ground or toss an enormous rock your way. This depends on where you are.

Summons in FF15 appear partially at random, but each has a trigger that sets them off. Titan is more likely to appear when your allies are downed. So long as you're in a space where he can appear the more of them that are downed the more likely it is his summon prompt will appear. When it does, hold L2/Left Trigger.

How to get and summon Ramuh
Ramuh will be made available to you pretty much immediately after the climactic battle with Titan at the end of Chapter 4 - most of Chapter 5 is about completing a set of tasks to bring him to your side. Mercifully you won't have to fight him.

Ramuh will come to your aid pretty much anywhere in the world - indoors or outdoors. This makes him one of the summons you're more likely to see and also one of the more useful of the astrals.

Summons in FF15 appear partially at random - the astrals being mercurial beings with minds of their own. Ramuh grows more and more likely to appear the longer a battle goes on. When his prompt appears, press and hold L2/Left Trigger.

How to get and summon Leviathan
Your next summon is the water god Leviathan, putting in perhaps its most memorable appearance in an FF game since it utterly failed to help you out of a dire situation in FF9. Leviathan will be made available as of the end of Chapter 9.

Leviathan is by far the least likely summon to appear, sadly - its water-based nature means that it will only show up when you're doing battle in an area close to large bodies of water

The summon condition for Leviathan is that Noctis be in Danger Status - when your health is out and the screen is tinted red. The longer you're in danger status, the more likely it is to appear - though that's a risky proposition, obviously. When the prompt pops, hit and hold L2 and Left Trigger.

How to get and summon Shiva
Shiva is the last of the main astrals that joins you, and will do so at the end of Chapter 12. When you head back in time, the ability to summon her travels with you, and like Ramuh she can be summoned pretty much anywhere, even indoors.

The conditions for Shiva to appear are a little more complicated - it's a combination of all of the conditions you see above.

So: party members knocked out, battles running long and Noctis in danger state. It doesn't have to be all three at once, but basically as these conditions stack up and last for longer, she becomes more likely to appear.