Is this note played at the same time as the regular sized note next to it? I think I remember my piano teacher saying something about notes that are only one step apart are usually written side-by-side and not above each other?
These are cue notes.
Depending on the musical context, these cues are either played or just a hint towards what's happening elsewhere in the ensemble. Other times they function as a "play them if you can" suggestion.
Based on the rhythms shown in this example, this looks to be a sheet music arrangement of a rock song, and are probably more in line with the "play them if you can." But you don't have to!
In any event, that small B♭ is to be played at the same time as the larger C. My guess is that, due to the different sizes of note heads, they gave a little extra space between them.