Can you start a sentence with the word And?

Yes. But be careful with it, because your writing can get that run-on sentence feeling if you do it too often. Usually the word AND will be used to continue a thought from the preceding sentence.

Some old-school grammarians still preach against it, but usage-wise, many good writers have done it, and it is perfectly acceptable. The most important thing is to avoid overuse. Make sure there is a purpose behind the choice and not just ignorance of sentence structure. If you keep those things in mind, it can be done well and to good effect.