Help me understand this book.

What do the period marks mean in the table?

I am not sure.

Can any of my brothers and sister help? 

Variations are numbered across the top. There are six on this page beginning with the Kieseritsky (col. 1) and ending with the Muzio (col. 6). I will write the variations rather than explain. You can then locate them on the chart and understand what the dots represent (they represent common moves between variations, but seeing is better than reading).

The Kieseritsky: 1.e4 e5 2.f4 exf4 3.Nf3 (Defining the "general" King's Gambit Knight Variation. The main other is the King's Gambit Bishop's Gambit 3.Bc4. I prefer the latter btw.) THEN under col.1 is to the left the number 4. That represent the defining and COMMON move 4 for the first three columns/variations. So move 4 for White in columns 1 through 3 is 4.h4. The fourth move in the Philidor Variation (column number 4 at the top) is 4.Bc4. So the 7th move for White in the Muzio Variation is listed as 7.e5. The lower-case letters represent variations for the listed "main lines. So 'c' or (c) as it appears on the page, is an alternative to Black's 8...Qe7. Let me know if you can now figure this out. I think this is called an ECO style variation crosstable. If you are serious about opening research this is more a beginning than an end. Databases wil hold more information, but they may not offer an assessment. Assessments are the +/= stuf at the end of the various lines above. Again, this page likely does NOT cover all the options that have already been played but are well tested lines in the other side of the coin.

As soon as I have time to study this I will reply here if I have any further questions. Thank you very much! 

Ok