*If you have already learned your basic scales this will help. If not make sure you do or this will not make sense.*
In the most basic of terms a mode is a scale within a scale. Hence why I said learn basic scales before continuing.
To be more specific let's take a look at the key of G Major (a basic understanding of key signatures is a must as well):
The G Major scale is spelt: G, A, B, C, D, E, F#, G. These are the notes commonly associated to the key of G Major. So if you are playing in G Major you are using some combination of the notes above. Ionian (or the first mode) is a major mode and mirrors the Major scale of a Major key.
For example to play in G Ionian you really aren't doing anything differently than playing a G Major scale over a G root note or G Major chord. When it becomes a different mode and adds a different characteristic is when compared to the root note or chord you are playing over.
The order of modes are: Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aoelian, Locrian.
To write these all out in the key of G they would look like this:
G Ionian: G, A, B, C, D, E, F#, G (Major mode)
Played over a G root note or a G major chord. G is your tonal center. THIS IS ALSO YOUR TYPICAL G MAJOR SCALE. Playing these notes over an A root note or A min chord brings us to Dorian.
A Dorian: A, B, C, D, E, F#, G, A (Minor mode)
Played over an A root note or an A min chord you will be playing in A Dorian. The only difference and what gives it a different sound is that you are playing over an A or A min and not G or G Major. That is what determines if you are playing the Dorian mode or the Ionian mode.
B Phrygian: B, C, D, E, F#, G, A, B (Minor mode)
When played over a B root note or a B min chord you are playing in B Phrygian. If played over a G root note or a G Major chord you will STILL be playing Ionian. If played over an A root note or an A min chord you will STILL be playing A Dorian.
C Lydian: C, D, E, F#, G, A, B, C (Major mode)
When played over a C root note or C Major chord this grouping of notes becomes the C Lydian mode. Again, if you are playing over a different root or chord even though you are playing the same notes you are playing a DIFFERENT mode!
D Mixolydian: D, E, F#, G, A, B, C, D (Major mode)
When played over a D root note or D Major chord this grouping of notes becomes the D Mixolydian mode.
E Aeolian: E, F#, G, A, B, C, D, E (Minor mode. It is also the Natural Minor Scale or the relative minor to the key of G Major).
Playing this series of notes over an E root note or E min chord will result in the Aoelian mode. To get an idea of what the Aoelian mode sounds like listen to Mr. Crowley by Randy Rhoads. Randy Rhoads used the Aoelian mode quite a bit.
Locrian: F#, G, A, B, C, D, E, F# (this is a diminished mode meaning it is basically a minor mode with the fifth scale degree flatted. In the key of G you would be playing the F# Locrian mode if playing over an F# root note or an F# dim (1, b3,b5) Chord)
Practice these groupings of notes over different chords and root notes. Notice how it isn't what you are playing that dictates the mode or the overall tonality as much as it is the ROOT note or chord you are playing over. Starting on A and playing the an A scale transposed to the key of G is NOT playing Dorian if you are NOT playing over an A root note or an A min chord! Don't worry about building up speed. Of course you can later but right now the important thing is that you get a grasp of how these different modes SOUND. Technique and speed will come later but focus on how they sound and WHY they sound the way they do and you will get much more out of this than just trying to blaze through and build up speed. Of course once you do have a firm understanding of this modes are a great way to build of speed later on.
Also notice that as you go through the modes each mode corresponds to a different scale degree. For example A Dorian is still basically the key of G Major but it is used for the minor second or A min chord with A min being the tonal center and so on throughout the key.