Is an accidental and a non-diatonic note different or are they the same? What if I play a non-diatonic chord in a key, is that considered an accidental?
An accidental is a sign placed next to a note in any key that modifies it away from the normal note of that key. Their usage can be a bit confusing since they don't always do the same thing in every key. For instance, a natural sign can either raise or lower a note by a semitone, while sharping an already flat note would raise it by a whole tone. Generally all notes written on a staff are based on the 7 natural notes, with the key signature modifying which notes are played instead of those notes (either a semi-tone or whole-tone above or below its natural note).
The sharps and flats that define a key signature are not considered accidentals, they are considered diatonic notes, or notes that are natural to that key. Therefore, accidentals are used to move to notes outside of that key. For example, if you were to play in Ab major then Bb would be considered diatonic to the key of Ab major, while writing in a B natural would incur an accidental. Conversely, in C major that same Bb would be written with an accidental while B natural would be diatonic.
So, diatonic would be any note or chord that belongs to the written key signature, while an accidental would be any note that is outside of that key. Accidentals are also used to return a note to its original quality, since accidentals only last within the measure they are written.
As an aside, the term "diatonic" is also used to refer to instruments that play primarily in a single key, like tin whistles and harmonicas. This is to contrast them with fully chromatic instruments that can play all of the notes outside of the key to which they are based. Diatonic instruments are understandably easier to design, produce, and learn to play. They also offer unique playing options, such as easy chord playing on a harmonica.
In a major key, using the appropriate key signature, a chromatic note will get an accidental, yes.
It's a little more complicated in a minor key - which can use the sharpened 7th of the harmonic minor scale, the sharpened 6th and 7th of the ascending melodic minor, or the accidental-free natural minor. All of these would be considered 'diatonic'.
A non-diatonic chord will INCLUDE notes with accidentals. It isn't, in itself, called an accidental.
Notes that do not belong to the given key signature of the piece will have an accidental in front of them in the music, regardless of whether the piece is major or minor. Chords that are made with accidentals are non-diatonic, but I have not heard them called accidental chords. They are usually named according to their function, be it a Neapolitan 2nd, a borrowed chord, a secondary dominant, what have you. Some non-diatonic chords contain more than one accidental, depending on how unrelated it is to the key of the piece.
Also, in the harmonic and melodic minor keys, the 6th and 7th notes are not "sharpened." They are raised a half-step. This is an important difference, as normally flattened notes will receive a natural sign as an accidental.