"I waited for three hours."
まちました (machimashita) is the polite past tense form of 待つ (まつ, to wait). まった is the informal past tense of this verb. As for ました (mashita), that is just the polite past tense conjugation of a verb. (書きました kakimashita, 食べました tabemashita). I'm certain there is no verb such as ました.
I don't know if duo explained this in the tips and notes as web version wasn't available last time I checked, but to form the polite version of a verb in present tense, change the last u to an i, then add masu. Examples Wakaru... Wakarimasu Hanasu... Hanashimasu Yomu... Yomimasu Matsu... Machimasu
Obviously you have to take into account that tsu becomes chi, and su becomes shi.
For verbs ending iru or eru (with some exceptions) just remove the last ru and add masu. Taberu... Tabemasu Miru... Mimasu
Then there are irregulars, like kuru becomes kimasu, suru becomes shimasu. For past tense, just use mashita instead. Hope this helped :)
However, when people speak they dont talk in kanji...no one stands there holding kanji flash cards up for you to read. You need to explore and understand the context to comprehend what is being said, this only comes with communication practice. Kanji doesn't exist when we speak.
I believe 時 (とき) is used to express the concept of time in general. Appending it to the end of a verb could be translated to something like "at the time of [verb]".
The major difference between 時 (じ) and 時間 (じかん) as counters is that 時 is used as a counter for the time (basically the hour part of the clock) and 時間 is used for a quantity of time in hours.