To solve this problem, ten schools founded the Football Association of England and made a set of standard rules in 1863. That was the year when moden soccer started.

(What does 'that' refer to in the last sentence? Does 'that' refer to '1863' or the sentnce before? Q2: What is the part of speech of 'standard, an adjective or a noun?)

Yes, "That" is referring to "1863."

In that sentence, "standard" is an adjective, describing the noun "rules."