更 (gèng) means "even more" or "-er". Put it before an adjective or a verb like 喜欢 to say that something goes one step further than something else.
Subject + 更 + adjective.
更 always comes directly before the adjective. It usually means the first thing already has the quality and the second has even more of it: if today is cold, tomorrow will be even colder. You can use 更 without 比 when it is clear what you are comparing with.
A + 比 + B + 更 + adjective.
A plain 比 sentence (他比我高) is already complete. Adding 更 makes it "even taller". The most common mistake is to use 很 instead: 他比我很高 is wrong. You will also hear 还 in the same place, with the same meaning: 他比我还高.
更 also goes before verbs of liking, such as 喜欢 and 想. 我更喜欢 is the usual way to say "I prefer". To ask which one someone prefers, use 你更喜欢 with 哪个 or 还是.
更 compares one thing with another: "more". 最 picks one out of a group: "the most". Both come before the adjective or verb, never after it, and neither is used together with 很: say 我更喜欢, not 我更很喜欢.
Practise this grammar point on paper: 20 exercises using only HSK 3 words, with an answer key on a separate page. Free to print and copy for your class.
Every example on this page uses only words up to HSK 3. Paste your own sentences into the level checker to see which words, if any, are above your level.