I read this here
Apostles can engage in battle against apostles and missionaries of other religions. Missionaries can only defend themselves, not attack, while apostles can both initiate combat and defend from other attacks. If your city is converted, your Apostles and Missionaries for that city will be for the new religion, so make sure you have enough Apostles handy to lead Inquisitions and clear out heresy in your own borders.
An Apostle needs to be expended only once using "Launch Inquisition" in order to unlock the ability to build Inquisitors. Unlike missionaries, they can perform theological combat. Like missionaries, they're relatively inexpensive. If you find yourself swarmed with another nation's Missionaries, hit them with a few Inquisitors until they GTFO. If that nation's Apostles are much stronger than yours, swarm them with Inquisitors rather than waste your Apostles.
The Mont. St. Michel Wonder gives all Apostles the Martyr ability, which creates a relic if that unit dies in theological combat. The only religious unit you should let die is an Apostle with the Martyr ability. Relics go in Temples, which you have to build anyway, and provide Tourism and Culture, offsetting the religious loss all nearby cities will suffer (bring your religion back by Removing Heresy with a few Inquisitors). The Stonehenge Wonder gives you a free Great Prophet, and Hagia Sophia lets your Missionaries and Apostles spread religion one extra time.
TL;DR: Inquisitors can be unlocked by expending an apostle to "launch inquisition" and are used for eliminating foreign religions from a city completely.
Inquisitors are weaker at theological combat than apostles especially if the Apostle has received a "Debater" promotion, which gives a +20 religious strength in theological combat.
Inquisitor base religious strength: 70
Apostle base religious strength: 110
Missionary base religious strength (can only defend): 100
This video explains religious warfare quite well
Missionary vs Inquisitor
Inquisitors 2k forums