What exactly is personal sin and communal sin?

Personal sin - Freely chosen actions that have negative effects on the sinners as individuals and on their relationships with other people.

and

Communal sin - Affects all people simply because we live in a community with one another.

Examples: Poverty, war, destruction of the environment.

What does it mean to say that Jesus understood the Kingdom of God as being communal in nature?

To say that Jesus understood the Kingdom of God as being communal in nature means that a relationship is more than God and individuals, rather, it is among individuals (person and person).