This is a pretty broad question but I'll tackle a couple ways ignite could be used best. First Ignite has 2 effects, many people forget about the second(go read), and each of these effects have best uses.
The damage over time part (50 + (20 * lvl) true damage):
This is good for ending someones life when they are low hp and may get out of range soon. This is bad when that someone has a shield or some other mechanism of avoiding damage while low hp (its true damage but you can still avoid it). Ex: Shen's Feint, Janna's Eye of Storm, barrier, etc.
This is good when you need/want a bit of extra damage going into a 1v1 trade (you don't always need to use it to finish people off). This is bad if you use it too early because then they will either just run away(leaving you without an ignite).
The damage received by Ignite is true damage, meaning it ignores armor/magic resists. This is good for a bit of extra damage on an enemy with armor/magic resist. It is bad when that enemy is a tank with a lot of hp because you may misjudge the actual life they have left, and Ignite may be better used on a squishier target.
Before moving onto the second portion you mentioned Tryndamere being a good choice to use ignite on, this is true, but not just because he's at low hp when Endless Rage ends and it will tick to kill him. It is due to the second effect, the heal reduction, reducing the amount he can heal from his Bloodlust. This leads to the second portion of the spell.
Reduces health regen and healing effects received by 50%:
This is good for when the other team has a healer or you're fighting someone with a self heal. Often times using ignite early on an adc against a team with Sona, Soraka, Nami, etc will help to win fights because the target will receive less heal. However, if the other team has no direct heals this portion of the ability will only affect Lifesteal,regen, and Spell vamp. In team fights as mentioned above this is useful to use on priority targets earlier.
Cleanse removes Ignite's damage, but the heal reduction remains. Keep this in mind when using it on a target with cleanse, as you'll only get half the benefits of this spell.
Ignite does not stack with other ignite's(the duration will just get refreshed), if you have a lot of ignites on your team already, it may be worth looking into other summoner spells.
Ignite does not play well with other heal reduction effects, meaning it doesn't stack with them. Miss Fortune's Impure Shots, Tristana's Explosive Shot, etc.
Fun fact: Tryndamere's full fury Bloodlust heal is 300 + 150% AP, Ignite's damage over time at max lvl is 410, so just remember without the heal reduction portion on Ignite, Tryn would most likely be able to survive.