"I am going to buy a desk the day after tomorrow."
No; both actions in the future would require the simple “ます“ form, only the past uses “ました“ (with ‘a’, not ‘e’).
“まして” is instead the polite “-て” form and had other uses, but it is not a tense per se, that is: it has no implications of past or future, it has a different function altogether.
I slipped up and typed です instead of ます: 明後日は机を買いです and it marked that as wrong (obviously), but provided the "correct" version of: 明後日は机を買う
Is that an error? I've typed that first sentence several times using masu instead of desu and gotten it correct every time. Not sure why I slipped today, but its now presenting the 2nd sentences as the correct answer. What does the う mean/do for 買い?
します is the conjugated, polite form of する, the word for "to do". (Though there are some set phrases like べんきょうします, where you can't directly translate it as "to do". You just have to remember these phrases as they are.) ます is what you add to a verb to get the polite form for future actions or habits. Like 食べます (I (will) eat).
ます has present nature and it is paired to verbs to make polite form (e.g. 食べます、飲みます）. でした is past form of です. So, if you want to say that you were a student you say "学生でした" instead of "学生です". Aside of that, masen combined with deshita used to make past negative form (-ませんでした）. So, if you want to say "there WAS NO chair", you write 椅子がありませんでした. CMIIW