I would offer an expansion on those answers concluding that it is not forbidden to start a sentence with "And." The examples thus far are all short sentences which are arguably suited to merging into a single sentence per Chris Browne, excepting the strong emphasis example offered by Sunshine.
My own frequent usage of "And" is associated with two contexts. The first is long sentences that do not easily bear further continuation, yet which are incomplete in their intended task. They require a further thought that, though necessarily connected, is sufficiently different to sensibly permit a new sentence. The "And" makes the connection of the two thoughts less burdensome than would be the case if all were combined into one sentence, however grammatically correct.
The second context is in speeches I write for politicians and business people. While there is obviously a distinction between what is fit for spoken versus written English, the gap need not be large between formal speeches and the written form. And so I say speeches should be written with a mind to being read more frequently than ever they will be spoken.
Now the "And" that starts the previous sentence can be seen as completely superfluous. A sharper writer would omit the entire entrance "And so I say," to achieve that vaunted goal of using the fewest words possible. But I do not acknowledge a linguistic famine that mandates such strict rationing.
I pose no defense against a charge of bad style. To me it is pleasing. And to my clients it has been effective. Or so it would seem by the fact that they return to the source of this bad style.
I can write compactly. I do, as required.
But, when the opportunity presents itself, I luxuriate in languid, listless, indulgent sentences, that succeed in communicating meaning, not for the miserly sake of communicating meaning, but to the more generous end of letting the language dance. And that can take the form of using conjunctions as instruments of pacing, or of bridges over which the waltz can move from one thought to the next.
This is what for me separates language as work from language as recreation.
I can do the work. But I want to play.