I don't have any professional training or background but I do "voices" every session. As a DM, I'll often have a half-dozen or so to do in a night. I don't bother with Joe Random NPCs (like barkeeps or blacksmiths... it's okay if they all sound the same) but named NPCs generally get something unique.
The trick is listening to the differences in accents. There's always a pattern to it. Certain vowel sounds change, certain consonants get emphasis or lightened. Once you learn to listen for what is different in an accent, it becomes possible to teach yourself to mentally "transform" on the fly.
For instance, Sean Connery. A HUGE part of his accent is that his sibilants aren't hard. "Yes" becomes "yesh" and "snake" becomes "shnake" while "absolutely" becomes "absholutely".
That's pretty much it for accents. For "voices", usually you also want to change volume, timber, and pitch. Having someone like Michael Jackson say "yesh" won't sound anything like Sean Connery. Michael had a soft voice that was pitched much higher than Sean's. Not everyone's throat is up to the challenge of modulating dramatically, but it's worth a try.
My point is the biggest, most important skill is learning to HEAR the differences. My greatest success was sitting down and deliberately learning Christopher Walken. It's an accent, a timber, and a choice of words that combine to make a unique individual. It took a couple days of watching some movies, but in the end it's a voice I can reproduce at will and sustain without drifting off into some other accent.
Someone else may have more professional input, but maybe this helps as a start.
Bam: Everything Anguish said. That's good stuff.
I have some voice acting training. I use voices often and have a proclivity for gnomes. (Most readers probably had a fit just then.) If you're just making a hobby of it, there's a few tricks you can do.
1. Practice your range. How low can your voice go. Start as deep as you can go and go as high as you can go. It doesn't have to be a song tone, just a natural voice tone.
2. Resonators. The main voice resonators are the chest, throat, mouth, teeth, nose, sinuses, and skull. Chest resonation leads to deeper booming voices and is often used by men for distances. The throat is probably where your normal speaking voice resonates, sex irrelevant, and for singing. The mouth is a loud, resonator without much force, like Mickey Mouse. (It's also the primary resonator for the Batman voice. Yes, I just tried it, sitting in front of my computer. Voice acting does that to you.) Very common cartoony resonator.
Teeth are sometime used by women and will make any voice sound smiley. Nose goes absent when you have a cold and makes anything sound flat. Teeth and nose are used often in french. Most people probably don't use their sinuses to any real degree, but those who can can do the Chewbacca grumbles.
The skull is a very loud resonator and reverberates higher pitches. Most high women's voices resonate here.
As you go through your range, place a hand over each resonator. You can feel the vibration move.
3. Mouth shape. The lips and tongue as well as the angle of your head over your throat can shape the sound that emerges. (Try this. Grumble a low note sustained, then tilt your head forward and backward. You'll hear changes. You may even stop making tone in some cases if you tilt your head back to far. Neat, huh?)
If you can control the shape of your throat and vocal cords to a high degree, you have a good deal of talent already and probably have the capability for a few impersonations. If you don't, the mouth will probably be where you have the highest degree of control.
Play with more O or E shapes when talking with a voice. Talk with your tongue hardly moving for a smaller voice. But of course, this leads to the next point:
4. Play. If you're just coming up with a voice for a hobby game and not looking to have it a huge part of your earnings, have fun with it. Go deeper, higher pitched, and you needn't go too cartoony, but play around with the voice and find something that fits. Impersonate people, or try to. Mix tones of voice with resonators. Find a tone and shape it around.
5. Saving a voice. When you find a voice, tie it to a certain phrase or sound effect. For example, if you've found a place where you can make you're voice crack and this is the perfect tone for someone's leadership feat, say, "Would you like fries with that?" over and over in that voice. The two would become interconnected and you can go back to a voice you've used before. This is very handy if you're impersonating voices. If you said, "I'M BATMAN" to get into the Batman voice, you've already done this.
6. Remember to breathe. Don't breathe at the neck or rib cage. Breathe with the belly. You'll get more air. And you'll probably need it if you aren't using your own voice for a period of time.
7. Verbal clutter and tics. Add a sigh. Click your tongue. Give a two-tone laugh like Jimmy Carr. Match these to character and don't mix 'em up. These can go a long way to fleshing out a character. Used too often and they may get stereotypical, but they will be recognizable. There's a reason sitcoms have catch-phrases.
In terms of accents, if you don't already have a basic accent impersonating technique, that would probably require a lot of time investment. Best recommendation I have would be to watch movie sources that use the accent and imitate them back into a voice recorder, and then listen to the recordings until you're no longer embarrassed by them. I will warn you, bad accents I react to viscerally. Oh, it hurts so bad!
I have to stop writing this now. There's more to say, of course. I have a book in mind, but it's more about doing sound effects. Mouth Sounds by Fred Newman. It comes with a CD.
My accent resources... I don't really have any that are for training purposes anymore. I did International Phoenetic Alphabet stuff so long ago it hardly registers. Not that I'd advise that, it's way too much work for what you're using it for.
Anywho, have fun!
Yeah actually. Limber up beforehand. It sounds silly, but muscle tension can inhibit vibration, and vibration is what you're after. Stretch things out, especially the shoulders and arms and the muscles on the side of your neck. If you can move better, your voice will carry better.
Otherwise, positioning has its greatest effect on the neck. Most of the changes will come from the mouth, lips and tongue. Anything else in the throat has to be done internally, which you may or may not have the talent to do. Like any muscle, training these things make's them more versatile and will have more brain-space connected to their use resulting in higher degrees of control.
Here, try this. I will assume you have the ability to cough. Cough while giving a sustained tone. Now, the cough constricts your throat resonator to a narrow point and closes it off. Now, try almost coughing, get close to fully closed and give your voice a strangled quality. With this new, narrow throat try talking. That's how I get to one of my old man voices.
WARNING! WARNING! You will want to warm up your voice before you try mucking around with this or you can lose your voice or otherwise cause discomfort. Say a few tongue twisters at a steadily increasing pace, sing something, (scales, anything), and give heavy voiced sighs for a period of at least five minutes. If you are in public, make sure you wave your hands a lot and jump up and down. (Sorry, a bit of the Gnome coming out there.)
Edit: OH! And drink water!
Minor speech impediments make for excellent tusked creatures. How do you talk if your teeth are too big for your face? Maybe your doubleyou's come out sounding wike Ewmur Fudd becawse yowu can't saiy dem wraight. Maybe you're a Naga and you hiss your Esses with an exssstra ssssbilant. Play with the consonants. Slide you're tongue across your teeth between a SSSSS and a THHHH. Form the W's in the back of the throat rather than with the lips for the Fudd speak. Heck, put pencils in your mouth and speak around them, then make that same voice when not having the pencils there, or at least a version of that voice you can maintain.
Watch Cartoons: Cartoon voices convey static things about a character that you may not realize and feed into subconscious stereotypes. They ARE their voice and every voice must be distinct. One of my favourites, Pinky and the Brain. One has a serious, deep voice based on Orson Welles. The other has the fudd-like speech impediment littered with nonsensical verbal clutter. You instantly know who's talking even if you're in another room. I could listen to a radio show with those characters and know what was going on. Imitate them, come up with your own variances.
Rolling Rs, Ls: Can you roll your R's? How about L's? How about you D's? (It's possible.) If not, don't worry about it. If you can, cool. At them in when necessary. This falls under lisping, accents, and verbal clutter and works well for Catfolk if you're stereotyping.
Plan ahead: How do they sound? If you have a sound in mind, you can create a character to accommodate. Otherwise, if you have a character and you need a voice, look at their facial structure and give them enough distinction to stand on their own.
Don't be too afraid to look silly: Experiment. Try to talk using only your lips and no tongue to make all the sounds. How many letters of the alphabet can you get through ventrilloquising using your tongue? These are just silly little ideas that might lead you to something you can talk as for a while. They're silly, but they warm you up. I've come up with characters that way, but the best way I make a character is:
Their Laugh: Very few people will laugh in the same way. A pixie will have no trouble laughing, but a serious, serene, necromantic sorceress may choke on her own laugh as it surprises her. I have a character (an actual character, not a character sheet) who's memory phrase is just his laugh.