A few notes:
1. Esterification can occur when sulfuric acid is not present. The reaction occurs slowly. Sulfuric acid in this case acts as a catalyst [it acts as a dehydrating agent, but also supplies H+ necessary for the reaction]. The process of esterification does not necessarily imply a catalyst is required.
2. Esterification is not confined to the reaction between a carboxylic acid and alcohol. It is correct in saying that an ester is a molecule that contains the COO group, but it should be made clear that an ester contains the
functional group COOR.
3. Synthesizing an ester from other reactants:
acid anhydride [not alcohol anhydride] + alcohol
acyl halide [RC(=O)X] + alcohol
carboxylate salt + alcohol
4. The reaction is a condensation reaction. However, the origin of the water molecule (where the OH and H comes from) is beyond the VCE syllabus. You should avoid talking about the reaction mechanism, simply stating 'a water molecule is removed' is enough. You should especially avoid talking about where the electrons in bonds go to, you're treading very dangerous waters there stating one is anionic and one is cationic. See
http://en.wikipedia.org/wiki/Fischer_esterification if you want a closer look at the reaction mechanism.
5. Somestimes some acid is left over, what times? which acid? [The catalyst or the carboxyl?]
You have given a good explanation, but instead of focusing on VCE level material, you have chose to focus on the reaction mechanism, which strictly speaking is not answering the question. Your explanation of the reaction mechanism is also technically incorrect, e.g. the OH in COOH is still called a hydroxyl group [not a hydroxide, which is the OH- ion], and by discussing these advanced materials in a slightly incorrect way, you have dug a hole for yourself.
My take on it is your teacher would have noticed you 'alcohol anhydride', and counted the rest of the paragraph as 'babble', hence marking it down.