Hello there! Very good questions. For most regular MBA programs you DO have to have work experience (usually between 3-5 years is the average). But there are a few programs that are designed for people right after your undergraduate studies. For example you have programs such as the HBS 2+2 program: http://www.hbs.edu/mba/admissions/2+2/, and the Yale Silver Scholars program: http://mba.yale.edu/MBA/admissions/apply/college_seniors.shtml
they are VERY competitive programs, of course
You also have a degree which is called Masters in Management (MiM) which you can get mostly from European Business schools. These programs are designed to be joined right after your studies. The most famous one is the LBS MiM, but many other programs are amazing as well.