An inkhorn term is a word that is unnecessary and used specifically to show off. It is particularly a word that is always unnecessary; misusing a potentially useful word does not qualify it as an inkhorn term. They are most cases of using a foreign word where an English one would do, or creating a new word when an old word will do.
An inkhorn is simply what we today would call an inkwell, and has been a associated with academics and scribblers for centuries. By the 1500s, the inkhorn had also become, to some extent, a symbol of the pedantic and affected scholar. Around this time scholars were working with a lot of languages; scholarly works were often written in French and Latin, there were many fascinating texts in Middle English to be translated and updated, and new scientific and technical words were being invented. Highly educated writers often co-opted words and phrases from foreign (and sometimes outright dead) languages or invented terms out of whole cloth. Critics often criticized them for this.
"I am of this opinion that our own tung should be written cleane and pure, unmixt and unmangeled with borowing of other tunges; wherein if we take not heed by tiim, ever borowing and never paying, she shall be fain to keep her house as bankrupt."
-Sir John Cheke (1514-1557), in a letter to Sir Thomas Hoby.
Sir Cheke was complaining specifically about the overuse of Greek, but he and others were also annoyed by the use of Latin, French, and any other language that did not come from good Germanic roots. Some critics passed from simply supporting Plain English and espoused using what we today would call Anglish, English based firmly around its Anglo-Saxon roots.
These days inkhorn terms might include anything from bon ami to floccinaucinihilipilification. Obviously, this problem has not been solved in the past 500 years, and will be with us for the next 500, for the simple reason that much of the population will always think that their hobby is more than deserving of specialized jargon, and that their maho shôjo anime is totally kawaii.