It is quite a shock, having spent all of my musical life in the landscape of the major and minor keys, to realise that they are only part of the picture.
The way to get in to that wider landscape is reasonably easy. The best way to start is to consider the C major scale (C to C on the piano keyboard, on the white notes). Running from C to C is the first mode, the Ionian. Run from D to D and you have just played the Dorian mode. E to E is the Phrygian, F to F is the Lydian, G to G the Mixolydian, A to A the Aeolian, and finally B to B the (seriously strange) Locrian. These scales (apart from the Ionian and the Aeolian) are likely to sound a little strange, incomplete somehow. But this is because the feel of these other modes are unfamiliar; once you get inside them, you begin to understand them and their distinctive characters are revealed.
The above is a quick way to get hold of the modes. But it is of course possible to play the Dorian, for example, starting from C or G etc. And to see how to do that, we need to get into just a little more technical detail.
Running from C to C involves a series of tone and half-tone steps in this order, TTHTTTH. If you were to play the D major scale (which contains an F sharp and a C sharp) you will notice it maintains the same pattern of tone and half-tone steps. The reason the F and C sharps come in is precisely to preserve that pattern of steps, which constitutes the major scale. What each mode does is cycle through this pattern, so the Dorian is constituted by the following series - THTTTHT (this differs from the major scale pattern in that a T has detatched itself from the beginning of the sequence and attached itself to the send). D to D on the white notes follows this pattern. And if you were to play the Dorian from C to C, you need to follow that sequence, which gives you the notes C, D, E flat, F, G, A, B flat, C.
The Phrygian follows the sequence HTTTHTT, the Lydian TTTHTTH, the Mixolydian TTHTTHT, the Aeolian THTTHTT, and the weird Locrian HTTHTTT (each time to generate the next mode, detach a step from the beginning of the sequence and stick it on the end).
Follow these recipes to play in any mode starting from any note. And enjoy!