If you can start the car there's an easy way to tell. Before you start the car loosen the positive battery cable to the point that you can easily remove it but it is till in solid contact with the battery post.
Start the car then remove the loosened battery cable. If the car keeps running your alternator is working but if the idle drops or the engine stops then your alternator is not putting out enough power to run the engine and needs to be replaced.
Either way, loosely replace the battery cable so you don't lose stereo presets, etc then turn off the car and tighten the battery cable and replace whichever component needs replaced, battery or alternator.
Also take a moment to inspect the battery posts and cable terminals and clean if necessary.