Modes are simply different ways to construct scales; instead of making alterations to key signatures with which you're already familiar, changes that make no rhyme or reason other than, "You do it this way to get this sound," you're taking key signatures you already know (because they're the key signatures for all 15 major scales) and applying them to different progressions.
A saxophone professor once explained the concept this way to me, and it made modes make perfect sense: there are seven modes, and each has a different, distinct tonal quality. Some are major, some are minor, and there's even a couple that are diminished. These tonal qualities can be ranked from brightest to darkest thus:
...with Lydian being the brightest mode and Locrian being the darkest.
I remember that order using the first letters as a word: LIMDAPL. I admit it's not the best mnemonic device, but it works for me.
The wonderful thing about modes is that you only need to know three things in order to be able to construct any mode at a moment's notice: all 15 key signatures (which you should know anyways), how to construct a circle of fifths (which you should know anyways), and the meaning of the non-word LIMDAPL (which I've explained above).
Let's say you want to construct a mode that started on D. To do so, you'd draw a circle, write "I" at 12:00, "L" at 1:00, "M" at 11:00, and continue writing the first letters of each of the modes at each of the subsequent hour marks, with the second "L" landing on 7:00, like so (forgive the crudeness):
Since you want to construct a mode starting on D, place D at the 12:00 spot, like so:
From here, say you want to make D Lydian: you'd move to that spot on the circle and adjust the key signature by the appropriate amount. In this case, you've moved one spot to the right, so you'd add one sharp to what D Ionian is. If you have three sharps, they are F#, C#, and G#, in that order. Strangely enough, those three sharps are also how you make A major/A Ionian. Therefore, D Lydian is a D scale in the key of A:
D E F# G# A B C# (D)
It works every single time. Say you want E Phyrygian: put E at 12:00, then move around the circle until you get to "P". Count how many steps you've made, then adjust the key signature: from Ionian to Phrygian is
I > M > D > A > P
That's four steps around the circle. To make the adjustment, subtract 4 sharps from E's key signature. E major only has four, so E Phrygian has no sharps and no flats; in other words, E Phrygian is an E scale with the C major key signature.
What about G Dorian? Ionian to Dorian is
I > M > D = two steps.
G only has one sharp, so you have to both subtract a sharp and add a flat. In other words:
G Ionian is a G scale in the key of G major
G Mixo-Lydian is a G scale in the key of C major
G Dorian is a G scale in the key of F major
Once you're able to do this on the fly, modes become a cinch.
As others have noted, modes are used often in jazz: when you're trying to solo, you're thinking about the chords as they change through the piece, which can be daunting. Using modes is a way to simplify the process.
Hopefully this was helpful. Just remember to add sharps/subtract flats and add flats/subtract sharps in the right order (order of flats is BEADGCF, order of sharps is FCGDAEB).
Let me know if I need to clarify something. Good luck!