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: 
Although the D in the scale of B will be natural in the key signature, it will often be sharpened when leading to the key note E. Think of the melodic minor scale for E. D will be sharpened on the way up but natural on the way down. 

So, if your chord is to resolve to the tonic E, the sharpened D will work nicely. You might find that a B minor chord works well if it goes somewhere else next.
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)