How can I be an effective support with Zenyatta?
Zenyatta does have a somewhat unique playstyle. His squishiness makes it so he needs to stay out of the open and stay near his tanks, preferably a Reinhardt who can use his barrier to shield Zenyatta and help make up for the squishiness.
Zenyatta's orb of Harmony heals 30 health per second, which is lower than Mercy's healing, but has a longer range, as an orb of Harmony will remain on an ally so long as Zenyatta has line of sight of that ally. One thing of importance is that Zenyatta cannot send his orb of Harmony through an enemy shield or barrier. Generally you want to throw this on whoever is taking the most damage, usually an offensive or tank hero.
For orb of Discord, this like orb of Harmony can be thrown at a range of 40 m, and remains at any distance provided that he keeps line of sight. Orb of Discord provides 50% bonus damage from all allies against the target with it on, so it is good to put on the enemy tank to help burn them down, or on an enemy offensive hero to make sure they die quickly before getting damage out. Note, losing line of sight for 3 seconds will remove the orb, and it cannot travel through enemy shields and barriers.
Zenyatta's regular attack does quite a bit of damage (35-175; 35 per orb in charged shot and 5 shots total; regular fire deals 45 per orb), especially if you have an orb of Discord on the target. His charge shot is great to use when ever someone gets close range and needs to be killed as quickly as possible such as a McCree or Reaper. His regular fire is great for wearing down the enemy from a distance, though it does have a travel time, so make sure to lead a little. Times when you should charge his alt fire would be, when McCree uses his ultimate Deadeye, when Reaper is Shadow Stepping at a destination, when Widowmaker is sniping in the distance as you want to not expose yourself to snipers, while Mei is in a Cryo-freeze (fire once she unfreezes), when Zarya's barrier is on someone (use it when shield wears off). You can also use it to help burst a Bastion or Torbjorn turret by charing in cover and popping out to unleash all 5 orbs at him.
His ulti heals for 100 health per second to nearby allies and makes him invulnerable to damage for 6 seconds. While using his ult, you can block a number of enemy ults from hitting your team such as Blocking enemy ultimates most notably: Reinhardt's Earthshatter, D.va's Self-Destruct, Mei's Blizzard (the line of sight starts with Mei's Blizzard Drone), Junkrat's Rip-Tire, McCree's Deadeye, Reaper's Deathblossom, Pharah's Barrage, Bastion's Tank Form, Roadhog's Whole Hog. The main uses for transcendence are for initiating strong pushes, to simply save yourself, to counter pushes done by the enemy, and to massively heal your team.
There's a few key things about playing Zenyatta:
1) Stay out of the fray. You have great range on all your Orb abilities (Harmony, Discord, and Destruction), and are very fragile. Keep near cover as much as possible, and get to it as soon as you're taking damage - you have no self heal save for your Ultimate, but do have shields which can regenerate two thirds of your maximum health. At the same time, try not to get too far from the team, lest you become easy prey for flankers like Tracer or Genji.
2) You should be aiming towards 100 percent uptime on your Orb of Harmony - if everyone is at full health, throw it on whoever is likely to take the most damage first when the fight happens. Your Orb of Discord should be up as much as possible, as it offers a large damage boost, but don't be afraid to be selfish with it - in almost all cases, it is better to throw it on someone that is trying to kill you, in order to drop them first. If noone is trying to kill you, then it should be on either your own target or the team's priority target (many characters in their ultimate, specific characters that need to be dropped for a push to work, or what have you - this is some tactical judgement you need to develop to get the best use of this ability). Be aware that abilities like Reaper's Wraith Form, Tracer's Recall, and Mei's Ice Block can clear off the Orb of Discord.
3) Manage everyone's health, and be as proactive as you can. Pocket Zenyatta is not a good plan; try to keep everyone in decent health, and make sure to Orb allies in Critical state so that they can heal up if they grab cover for a few seconds. This also helps charge your ultimate very fast.
4) Your attack is quite effective when combined with your Orb of Discord. A fully charged alt-fire on a Discorded Bastion is a kill if all five orbs hit. The charge attack is good if you're ducking in and out of cover, or waiting out an opponent's ability duration, but otherwise stick to the primary fire. Precision is rewarded here, so aim for the head.
5) Your ultimate is very powerful, but does not "block" an enemy ultimate. Rather, it heals through anything less than a single hit that removes a character's entire health pool. Thus, this protects your team from abilities like Death Blossom and Rocket Barrage (because you out-heal the damage), but not things like RIP-Tire or Self Destruct. Alternatively, you can turn a fight around by using it when your team is at low health, or to drive a push through, or to help remove an especially sticky Bastion or Turret... It's extremely flexible, but more importantly it charges fast, so you can be relatively liberal in using it.
6) Remember that you are the most fragile character in the game; even Tracer is harder to kill. Wherever possible, keep to cover and poke in and out to refresh your line of sight for your orbs. Engaging weaker characters or close quarters characters from range can be effective as well.