In an empty universe, except for two equally massed stars, how would they orbit? Or, for another example, if the earth suddenly grew to be the mass of the sun, how would they orbit, or interact? Would they simple pull together and combine? In the earth example, because the earth is already moving, would inertia keep it orbiting the sun?
The orbits would be elliptical. What kind of elliptical depends on the distance of the stars from each other and their velocities at a given point.
At any given time, if you draw a straight line between the two stars, that line will pass through the barycenter of the system (the center of mass). The stars will move more slowly when they are farther from the barycenter than they will when they are close to it. If you increase their speed at periapsis (closest approach to the barycenter), they will get further away from the barycenter on the other side and their orbits would be more elliptical.
The earth would migrate inwards toward the sun if it suddenly grew in mass. If it grew close to the mass of the sun, the sun and newly-gigantic earth might become a binary system, where they orbit each other.
Though all objects in orbit "fall" towards each other at all times, they move out of the way of each other fast enough to not collide. If the earth gains more mass, the gravitational pull between it and the sun will become stronger. As long as the process by which it gained mass kept the momentum proportional (ie. it keeps the same velocity), it will no longer be "getting out of the way" of the sun as fast as it falls towards it. It might not fall into the sun, but the result would be that it would fall closer to the sun and wind up in a more elliptical orbit. If became just as massive as the sun, both suns would be suddenly drawn together very strongly. I suspect they would collide, but I haven't done the math on that.
Take a look at Two body problem article at wikipedia. You'll find how to calculate any two body orbit given the mass of the bodies, their initial positions and initial velocities.
The orbit would not necessarily be elliptical, it could also be circular, parabolic, hyperbolic, or radial. There are two parameters which determine the shape of an orbit, the angular momentum, and the orbital energy. If the angular momentum is zero, the orbit will be radial.
Low energy orbits are bound and periodic, the shape is either a line, a circle or an ellipse. Medium energy orbits (parabolic) are non-periodic, the shape is either a line, or a parabola. High energy orbits (hyperbolic) are escape orbits, they are non-periodic, the shape is either a line, or a hyperbola.