I don't fully understand Newton's second law of motion, F=MA. What happens to an object in outer space when a force is applied? Does it remain at a constant speed? Or, does it continue to accelerate, thus going faster and even faster? I've reviewed several Youtube videos explaining that the object will continue forever. Yes, but at what speed? Thanks.

First, the correct formula is \sum \vec F = m \vec a. Note the lower case 'm' and 'a' for mass and acceleration, respectively, and also that the left hand side is the sum of all forces acting on the object. Force and acceleration are vectors, which means they have both magnitude and direction.

To answer your question: if a constant force is applied to a mass and there are no other opposing forces it will accelerate as long as that force is applied. Because it is accelerating, it is gaining speed with every passing second. If the force is removed the object will continue at the same speed until some other force causes it to change velocity. There is one complication to this that Newton didn't comprehend, and that is that the speed of light is the universal speed limit - nothing can go faster than that. But for everyday objects moving at "reasonable" non-relativistic speeds, Newton's formula works quite well.