I have heard it explained that every year we are evaluated as though in a court, our merits versus our transgression. On Rosh Hashanah we do not concern ourselves with the Teshuvah but concentrate our intention on 'making Hashem our King'. The concept of Malchus, or Kingship, is the main thought for Rosh Hashanah. We say "May you be inscribed in the book of Life" to our friends because we desire that we all be granted another complete year of life in this world.
The book of life is the book of action because while we are alive we are capable of doing things. Our actions in this world, specifically the doing of mitzvot, are one of the reasons Hashem grants us a year of life. But most of us also do a lot of transgressions in a year, and our doing of mitzvot sometimes becomes stale and routine. This is why the week between Rosh Hashanah and Yom Kippur are the best time to do complete Teshuva.
We must each evaluate our middot {character traits} and identify the bad traits in order to try to rectify them. We also must attempt to right any wrongs we have made between ourselves and our 'brothers' and 'sisters' and our 'neighbors'. We do this by asking them to forgive what we did wrong. Those transgressions against Hashem we must introspect on, and attempt to modify our thoughts so that we should never transgress again.
The decision is 'sealed' on Yom Yippur, that is the decision of whether we deserve another year of life is made on Yom Kippur. Although it is also true that this decision can be repealed until Hoshana Rabba.
http://njop.org/resources/holidays/complete-guide-to-holidays/hoshana-raba/
3) While G-d judges the world on Rosh Hashana and concludes the verdict on Yom Kippur, on Hoshana Raba the verdict receives its final seal. One therefore has time to complete the teshuvah, repentance process, up until the closing hours of Hoshana Raba.
Cheshbon HaNefesh: (lit. “an account of the soul”); a process of stocktaking and introspection with regard to one’s Divine service
http://www.chabad.org/holidays/JewishNewYear/template_cdo/aid/971407/jewish/Soul-Accounting-in-5-Steps.htm
If we have properly gone through this process of self evaluation {Cheshbon HaNefesh} then we merit being allotted another year and thus be inscribed in the Book of Life, the book of action in this world, the ability to do mitzvot.