Can you explain "Visits and Unique Visits?" 

A visit is a group of page requests identified by the same IP address and browser within a period of time (configurable in the config screen), not identified as a bot.

So if you have 10 people each requesting 5 pages within 15 minutes 3 times a day you will have:
5 pages within 15 minutes (default) = 1 visit
3 times a day by 10 people = 30 visits a day
However there are only 10 different combinations of IP address and browser so you will have 10 unique visitors.

If the 2nd day you have 5 people that have visited your site yesterday and 5 people that never visited your site, you will have 10 visits for that day, with 10 unique visitors.

The unique visitors value is depending on the period of time you selected. If you now switch from day view to month view you will have:
20 visitors
15 unique visitors.

To have an indication how many people are revisiting the site we added the 'Visits Average'. This is visits  divided by unique visits. So if the value is 2 than the average visitors are visiting your site twice (the selected period)

What is "Visits average"?

Visits average will show you how many visits are new visitors or visitors that previously visited your site. If the value=1, then all visitors are unique. If the value=2, then all visitors (on an average) visited your site twice within the selected period. 

Are the visits always originated from the displayed IP address?

From version 3.0.4.715_dev ther is implemented a new feature to recognize visitors also by using a cookie.

Using this cookie feature, the visit is counted for the IP/Client combination, which was used during the first visit. If the IP or the Useragent string is going to change, it would not reflect the newest values on screen.

Why are the total unique visitors of all days more than the total unique visitors for that month in the year table?

The total unique visitor (tuv) in the year table is less or equal to the tuv in the month table. This is due to the larger period you check for double users.

Example:

Month table:
day1: visitors: 3-2-2-3-1-3
day2: visitors: 4-2-2-3-4-3-1
uv day1: 3
uv day2: 4
tuv --> 7

Year table:
month_x: visitors: 3-2-2-3-1-3+4-2-2-3-4-3-1
uv --> 4
tuv --> (will also be 3 less)

That's why there is no total unique visitors counter in the month table, since it doesn't say anything that makes sense...

From version 2.1.2 there the value in the totals bar will be the real summery from the database and not the sum of the above numbers.