"This is mom, and this is Aunt."
Translation:Це мама, а це тітка.
How do we know the difference between the various "and" words such as "i," "a," and "ta?"
і and та are mostly interchangeable. We have both for euphony. For example you wouldn't want to use i before a word that also starts from i as it would be hard to say. Note, that та also can be used as a synonym to але (but). A is used when you want to express contrast: Це яблуко зелене, а це - жовте (This apple is green, and this one is yellow).
So, Ukrainian has also a cursive form for the letters? Because I guess the third word there is not ma...
Yes, you are right, I wanted to put emphasis and didn't realise it completely changed the appearance of the letter. You can check out the cursive alphabet here: http://toloka.hurtom.com/photos/140918104052101836_f0_0.jpg