There are many different types of synesthesia that one can experience (and in many cases people do not literally project these experiences--that is, experience them as though they were external sensory stimuli), but as far as I have ever been able to tell, there are only two that I do have.
First is what's called grapheme to color synesthesia. Letters of the alphabet, for instance, have colors that are intrinsic properties according to my perception. I do not literally SEE the colors on the page as I write, but I can get a sort of mind's-eye superimposed image as I look at them. I do not will this...it just happens. This type of synesthesia is one that can be experimentally proven online, BTW, without having to do a scan for anomalous neurological activity. The way the test is done is to give the subject a color palette like the one you see in Photoshop (the very detailed one with the slider you can move on both the X and Y axis) and put up letters and numbers in random order and ask the person taking the test to select the color they see with that letter, and to run through this two or three times. A non-synesthete is not going to have the response time or accuracy that a synesthete will. A synesthete's response to the test is going to be WAY on the far end of the Bell curve, and when I did it I most definitely hit it beyond the 99% range. (Can't remember the exact number, but it was high enough that there was no statistically significant doubt of the result.)
This can make doing things like picking out typos or picking numbers and other data out of long lists very easy for me to do--if I focus on the colors rather than the meanings of what I'm seeing, I can make what I want to see stand out very brightly indeed.
My other form of synesthetic perception is sound-to-sight. Now for most synesthetes this manifests itself as pitch-to-color, or even entire songs having a "color." For me, it's all in greyscale with absolutely no color information (which actually allows me to understand what a person without the ability to perceive color might find interesting or beautiful...and I also suspect it gives me a very odd perspective on what music does and doesn't sound good). I perceive the individual sounds that make up a song...I can pick out each instrument and see information about its timbre and pitch (though NOT with the precision of those who get color information). I can even visually perceive the use of certain effects like high/lowpass filter, phasing, flanging, clipping, and other uses of sound, as well as look at the acoustics where a sound was made (reverb, frequency range, and so forth all become very obvious to my eyes). It's also easy for me to isolate parts of a harmony because I can visually track each instrument.
MOST of the time I do not actually see this information in front of me, but certain loud sounds that startle me have occasionally caused me to see a white flash. (At night with my eyes closed I can see it more easily, but on very rare occasions it's been bright enough during the day to see it. MUCH more frequently, though, sounds give me the physical feeling in my eyes that they are reacting to light, but there is no external perception of light.) That said, my listening process works differently than most people's in that if I REALLY want all the detail, to truly immerse myself, I hear through my ears but listen with my eyes. When I really want to listen to music or any other sound the most fully, I close my eyes so that I can track the images and actually let my eyes go towards what I want to listen to. They move so fast that external visual input can make me lose a lot of the detail.
I experience the images as an intrinsic property of the sounds...which has made me give up every single time I try to learn how to read music, because no matter how much I try to get it through my head, it is not RIGHT as a visual representation of music. Unlike the more common synesthete with sound-to-sight, I don't think my abilities would ever help me to master an instrument...while I am reasonably sensitive to pitch, I don't think I'm quite enough so to reach really high levels without being able to read music. That said, I think I COULD learn to mix and produce music that others composed very well, because I can see what is and isn't good composition from that perspective, and once I learned what control did what, I can easily picture myself closing my eyes and "feeling" my way around the controls as I listened, to bring it into balance.