Correct spelling, explanation: conductor is a word based on the verb to conduct and the suffix -or. Now, in this case, the suffixation gets a little complicated, because suffixes -or, -er, and -ar carry the same meaning. In the case of conductor, the base word is conduct, and based on unwritten rules of suffixation, words ending in ct generally get the or ending. Just think about words such as act and actor made on this basis.

Definition of conductor:
noun
1. a person who collects fares on public transport
I caught the last bus, had no money, and I was unlucky enough to meet the conductor so now I have to pay a penalty.
2. the person who guides the orchestra’s performance
I always admired how a conductor can lead so many instruments with just a baton and his movement.
3. (scientific) a material that allows for a flow of current.
Water is a quite good current conductor which is why electricity in water is so dangerous.

Incorrect spelling, explanation: word conducter is incorrect, but you may wonder why, as the suffix, -er is sometimes used in similar words. Well, when it comes to choosing between -er, -or and ar, the most important thing is the two or three last letters of the base word. In the word conduct, the two last letters are ct, in which case, you should use the -or suffix. Meaning, it should be conductor, not conducter. So, if you ever wonder how to spell conductor, think of similar words, such as actor, to guide you.