I bought a brand new battery. I don't drive the car daily. After sitting for about a week, the car doesn't start. What can this be? And I have checked the alternator.

 

What year Monte?
I assume the car needs to be jumped, then it runs fine as long as you drive it daily?
Sounds like a parasitic drain. Typically most people begin with "do you have any aftermarket electronics, like lights, radios, speaker amps". Sometimes if those are not wired in properly, they end up staying on and draining the battery.
I can also tell you from experience, it can be the alternator. My '04 had a parasitic drain that took a long time to find because it was only driven once in a while during summers only. I pulled the alt and had it benched tested numerous times at multiple locations. Only ONE test attempt failed, re-tested on the same bench and it passed. Had it also tested once or twice on the vehicle. I finally replaced it when it began putting off a smell of burning electronics (and I had multiple people near the car notice the smell). After I changed it, no more problems.

There is test equipment to help you find a parasitic drain, I don't have that available to me.