Question: I find when playing in a minor key I always end up playing the major version of the V chord. (So in Em I always end up playing a "B major" chord.)

However, the third of this chord should be played as a minor when looking at the key and it seems like using an accidental as frequently as I do is not really in the spirit of accidentals.

Am I making a mistake in my music theory? Am I playing the key I think I am? Is this just a mode? Do I need to start playing more B minors?

Please bear in mind I am relatively new to music theory and would appreciate it if you treat me like an idiot. Thanks.

Answer: 
Two points.

First: the scale of a key or mode is a framework. It doesn't instruct you NOT to use other notes. A piece of music that used ONLY diatonic (in the scale) notes and chords would be very basic and possibly very boring (although there's also beauty in simplicity).

Second: in traditional harmony the dominant chord of both a major or minor key is normally major. If we're writing the sort of music that deals in dominant > tonic relationships (and that's the bulk of Western music for the last few hundred years) the dominant chord NEEDS to tend toward the tonic chord. A large part of that tension comes from it including the 'leading note', the note a semitone below the tonic.

We don't HAVE to use major dominant chords. But we don't have to reject then either. You're doing nothing 'wrong' by sharpening that 7th into a leading note. Equally, you're doing nothing 'wrong' by investigating the sound of a minor dominant chord.

"Theory describes, it does not command".
Say we play in Em (relative major is G, one sharp). The V chord of Em should be Bm (B D F#). Although playing B major adds a sharp to D it is not considered as an "error" since it can naturaly lead to Em. It's clearer if you play B7 (B D# F# A) as it's the dominant chord of Em.

Sometimes the II chord of the major scale (Am in G) can also be played as a II7 to lead to the V chord. Even the I of the minor scale can be «dominant»-ized.

Try to play those progressions:

* G Bm Em Em Am D7 G G (nothing special)
* G B7 Em Em Am D7 G G (B7 leads to Em in a strong way)
* G B7 Em E7 Am D7 G G (same with E7 to Am)
* G B7 Em E7 A7 D7 G G (same with A7 to D7)