Translation:I did not stand up.
たち is the "masu form" of たつ, which is the infinitive form. Basically, you MUST conjugate たつ into たち in order to attach it to ます, ました, ましょう, etc. たちます is correct. たつます is incorrect. So, they mean the same thing, they're just different forms of the same verb. Also, どっち means "which" and どこ means "where." They aren't related, other than both of them being question words.
tachi is base two of the verb tatsu. To get the masu form of the verb tatsu you take base 2 tachi and add masu to the end. Base 2 can also be left as is, as a verbal derived noun. docchi is a contraction of dochira and means which (way). Dore means which one and is the question equivalent of kore, sore, are - usually used to inquire after directions, dochira can also be used as a polite equivalent for dore.
It's a rule for conjugating - verbs are made up of stem, base (there are 7 bases) and verb ending. The first 5 bases follow the order of the ranks of kana that we learn so for 立つ for instance its first 5 bases are た、ち、つ、て、と（う） - each base has a different purpose/s. Just briefly here are a few examples - stem + base 1 + negative ending 立たない - present negative active plain form - (I) do not stand
stem + base 2 立ち - noun or can add ～ます ending to get present active polite form (I) stand
stem + base 3 立つ dictionary form (the form that you will find a verb in when you look it up in a dictionary), also present active plain form - (I) stand. Japanese verbs are divided into 3 groups ichidan, godan and irregular - most verbs are godan - 立つ is godan and they all follow the rank order of kana. The only thing you have to remember with this is that verbs ending in ~う in plain form (ie. dictionary form or stem + base 3) have わ as their base 1. Example あらう - to wash, 洗わない - (I) do not wash