The mitzvah (commandment) of taking lulav (palm branch) and etrog (citron) applies, according to the Torah, only on the first day of the holiday. As the Pasuk (verse) says, Ulekachtem lachem bayom harishon: Take these four species on the first day. Inside the Beit HaMikdash (Temple), when it stood, there was a mitzvah to do it for seven days, because the Pasuk says, U'semachtem lifnei Adonai Elohechem shiv'at yamim: Rejoice before God for seven days.
Without a Beit HaMikdash, by the law of the Torah, we only have to do the lulav and etrog on the first day. However, the Hachamim (our elders) instituted that we continue to shake lulav and etrog for the remaining six days to remember the days of the Beit HaMikdash, so that we don't lose the tradition of what was done in the time of the Beit HaMikdash. But that is only mid'rabbanan (from the rabbis). We only have a requirement on those days to shake it from the rabbis. We still make a Beracha (blessing), because there's a mitzvah to listen to the rabbis. As we read in the Pasuk yesterday, “Ask your ancestors and your elders, and they will tell you.” So when the Hachamim, our elders, tell us what to do, that's like a mitzvah from the Torah. Therefore, we have an obligation to listen to them, and we make a Beracha.
But the fact that the remaining six days are only mid'rabbanan lowers slightly the minimum standards for fulfilling your obligation. So, two examples of this. On the first day, the person has to have full ownership over the lulav and etrog. He cannot borrow it from somebody, and he cannot be a partner in the lulav and etrog. But for the remaining six days, if he borrows it from somebody, he fulfills his obligation. Secondly, we know that if you have an etrog whose pitom (stem) falls off, it's not a kosher (fit for ritual use) etrog for the first day mide'oraita (by the law of the Torah). But for the remaining six days, it's considered kosher, and a person could use it. So because the standard lowers from de'oraita to derabbanan, some of the laws themselves are also lowered as far as the minimum standard to fulfill.