You don't say if you're playing with a band using a house PA or are playing solo so my answer is going to be a little broad, but maybe that's good for the other folks. I'm not going to put any links into this answer because you can google all of these phrases to find what you need. Teh Internetz luvs to sell stuff to musicians.
A DI unit, DI box, direct box, or simply DI (variously claimed to stand for direct input, direct injection or direct interface), is a device typically used in recording studios to connect a high-impedance, line level, unbalanced output signal to a low-impedance microphone level balanced input, usually via XLR connector. DIs are frequently used to connect an electric guitar or electric bass to a mixing console's microphone input. The DI performs level matching, balancing, and either active buffering or passive impedance matching/impedance bridging to minimize noise, distortion, and ground loops.
DIs are mainly used when you're plugging into a venue's ("house") sound system ("PA" - Public Address). You don't need to worry too much about what high impendance or low impedance means. Just remember that if you're going to plug a quarter-inch electric guitar cord into something with an XLR socket, you're going to need a DI.
XLR (mic) connectors:
A mixer is what a sound engineer uses for mixing and tweaking the sound sources or channels. A mini-mixer usually refers to a mixer that has 4 or less channels.
You can always play with just a mic on a boom stand that plugs into the house PA. You don't need a DI for this, because the mic is already XLR. There are a huge variety of mics so I'm not even going to go into that. Any mic you use will have an XLR plug so you won't need a DI.
A combo amp is a single unit that has a mini mixer, power amplifier and speaker ("driver") in the same box. Combo amps are typically used by electric keyboard and guitar players and typically take an electric guitar cord as input, although some also have XLR inputs.
An alternative to a combo amp is a powered speaker. A powered speaker has a power amplifier and a driver in the same unit. They typically have one or more XLR inputs. Powered speakers have better frequency response than guitar combo amps but are more expensive and typically require a mixer as well.
If you're playing solo you will need at least a combo amp but you'll probably also want a mini mixer so that you can tweak your EQ (many bands of frequencies that break down to bass, mids and treble).
The big range comes when you start talking about violin pickups. There are, as you've noted, a large variety of choices here.
If you decide to go with a transducer, then it will be installed in, under or on the bridge. Have a tech install this for you. If you use a piezo transducer you will need a preamp and if you go electric, you will need a DI unless you plug into a combo amp.
An electric pickup will only work if you have steel strings on your violin. This will also require a tech to install it for you unless you're comfortable, uh, fiddling with your violin.
If you want to minimize changing your violin--which may affect the tone--then you may want to consider a clip-on horn mic. You attach the clip on to your chin rest or to a little piece of plastic that you glue to your chin rest. This is the solution that the fiddler in our band uses.
The last choice is whether to go with a cord that plugs into the pickup or go wireless. A cord is cheaper but it definitely affects the balance of your violin and for you may affect your ergonomics. Most of the professional fiddlers I know who don't use a mic use a wireless setup. The fiddler in our band has a wireless clip on horn mic. Wireless mics transmit radio signals to a radio receiver that you plug an XLR cord into which then goes into the PA.
Hope this helps.
Please comment for any additional questions and I'll edit my answer here.