All there in the title. If a second moon appeared in our sky, what would happen to our planet?
"It starts with an earthquake, birds and planes..." No it's not quite that bad but there will be minor near term seismic adjustments as the current tidal equilibria all change. We're going to have to remeasure all the tidal ranges for every beach, bay, and cove on the planet and work out new tidal schedule and projection formulas. Mayflies and other creatures that either spawn or migrate according to the existing Lunar cycle, either because of lighting or tidal effects, probably go extinct in short order in all the confusion. Depending on the composition of the new moon it could effect the Earth's Magnetosphere, goodbye most long range migratory bird species too. Sea life is going to freak at a minimum there could be extinctions there too. Depending on relative orbital positioning of moon 1 and moon 2 tides may double, or more causing salt water inundation of large areas of the coast.
That's all I can think of off the top of my head that doesn't involve them hitting either each other or the Earth due to orbital mishaps.
I'm going to ignore what would happen if a second moon suddenly appeared in our sky. If that's actually your question, edit your question and I'll modify my answer. I'm going to assume, "what would a world like ours be like if the only difference were a second moon of the same size?"
If the two moons were in the same orbit
Twice the Tides
Most likely two moons sharing the same orbit would be 180° apart. In this case, tidal shifts would occur twice as often. I would expect the size of each tidal swing to be the same as our world since the mass of the earth is so much greater than that of either moon, the effect of the moon on the far side of the planet would be nothing compared to the one on the near side of the planet ("near side" being the side where we're measuring tidal effects).
Twice the Erosion
However, with twice the tidal swings comes twice the erosion. Consequently, your world would have more beaches with finer sand. Cliff areas would be more deeply undercut. Etc.
A more homogenous ocean
This would also affect currents. The water isn't being pulled inland and then away again so much as it is being dragged along by the moon. This dragging affect is one component in what causes currents (heat from sunlight and depth of the oceans both likely contribute more, you'd need to consult an oceanologist for more specific data). However, with the drag occuring twice as often I can imagine the mixing effect to homogonize the ocean more than it is, reducing the strength of currents running contrary to the orbit of the moons and strengthening those that run with them.
Shipping would love it
Even today, shipping loves to work with the tides to get them where they're going efficiently. Giving them twice the number of opportunities simply means greater departure and arrival time flexibility.
Calmer, uglier weather
I think, I'm not sure, but I think this would have a substantial calming effect on weather in general. Increasing frequency tends to reduce the effects of peak energy — but it also means a greater root-mean-square (RMS) energy. So, and I'm pretty much completely guessing here, storms would be less frequent, but when they happen, they'd be more virulent. (The reason I'm guessing is that, like currents, weather is predominantly affected by solar heat, which isn't changing.)
If the two moons are not in the same orbit
From an orbital mechanics perspective, it's more likely that the two moons are not in the same orbit. It's very, very unlikely the two moons would be locked in their orbits, meaning the time for moon A to orbit is different than the time required for moon B to orbit.
When the two moons are lined up, the tides are at their maximum. At apahelion (furthest point from the planet) the tide would be greater (higher) than the single moon can produe. At perihelion (closest point to the planet) the tide would be much greater (higher) than a single moon could produce. This would force people to build further away from beaches and the beaches themselves would be much deeper than we see here on earth.
Shipping would be a pain
Where the previous example would be a boon to shipping, this would be a bane. Shipping schedules would become quite complicated with, possibly, the perihelion tide avoided where possible (think of the ocean swells that sucker could produce, hope you're traveling with the orbit my friend).
Weather fronts would be created by the perihelion tide
Enough energy is dragging the ocean at perihelion that I can imagine (with exactly the same level of confidence I had with my last weather prediction) that weather fronts would be created. The tidal effect would "push" the weather along with it. This would, I think, create predictable storms (and probably nasty suckers) during the perihelion sweep. I could be wrong, but it would make a good story!
Depends on distance
Just like 1 moon has an effect on our tides, so too would a second. When these two moons align would surely result in a bigger tide causing flooding in areas not use to such levels of water.
It could also throw our current moon hurling out into the solar system. Or they could collide with eachother destroying our planet in the aftermath.
Also, depending on distance, could steal atmosphere from us.
Distance and mass are the keys to determining gravitational force