Real Sound Blasters don't need a driver to initialise or support them. Later PNP Sound Blasters (SB16/AWE) and clones may need one-time initialisation. Exotic cards may need memory-resident translation layers.
Games use a collection of per-card drivers to 'talk' to the appropriate hardware interfaces. These can be hard-coded into the game, or a collection of external files like in HMI or Miles.
Depending on the game and card you have, one or both of these may apply.
To play Halloween Harry on a real Sound Blaster ISA card, no extra drivers are needed. The SB support is hard-coded.
To play Theme Hospital on a SB Live! PCI under MS-DOS, two different kinds of drivers are used. The SB Live! PCI itself requires a Creative resident MS-DOS driver to imitate the hardware interface of an MS-DOS-era SB16 card. The game itself uses a third-party Miles driver to abstract the Sound Blaster (or Pro Audio Spectrum, Windows Sound System, Wavjammer...) card for the game's use - in this case you'd pick Sound Blaster 16 on the setup screen.
All the answers given here so far are correct, for different scenarios. What may confuse is that there are two distinct phases that both could correctly be called 'drivers'. Let me describe what I mean:
Pretty much everything here applies to both digitised audio output and AdLib/OPL2/OPL3 support equally, but I'll focus on digitised audio.
1) Initialisation and support to -provide- the hardware interface
Legitimate, first-party Sound Blaster series cards are programmed directly through I/O ports. There's a chip on-board known as the 'DSP'* which handles all the data movement between system memory (requested through a system DMA channel) the card's DAC/ADC. If you have a real Sound Blaster card, and the game is hardcoded to 'talk Sound Blaster'** then that's all that's needed.
If you have a clone card, a third-party 'compatible card', or a later SB16/AWE card then one of the following applies:
- The card exactly mimics the behaviour of one the Sound Blaster series cards and no further intervention by a 'driver' is necessary. The 'Snark Barker' is designed as an SB 1.x clone - it transparently acts as an SB without a driver.
- The card starts off 'inert' and requires some initialisation. This is common with 1995-1997 PnP compatible cards whose IRQ and DMA settings are done in software rather than by jumpers. My Avance ALS100+ based card and CMI8330 cards require a start-up program to be run before they will work. This program talks to the card, tells it what IRQ and DMA to use, and from that point onwards the card acts as a clone card. No persistent program resides in memory to translate a game's Sound Blaster DSP commands into Avance commands, etc. If you've installed the 'driver' for a clone-like card, this most likely applies to you.
- If the card is not capable of directly acting as a clone card because it is exotic like a Gravis Ultrasound, or a very new (relatively speaking: post 1996) Sound Blaster / Ensoniq PCI card, then it cannot be simply initialised into acting as an SB clone card. These cards require a software shim layer to be loaded resident to intercept Sound Blaster DSP commands and translate them in real time to commands the card understands. For the GUS, this is SBOS. If the game you're playing natively supports GUS, then you don't need SBOS. For cards without FM chips/clone chips present, the shim layer may synthesize the audio in software in real-time, with mixed results. For Ensoniq/SB Live/SB Audigy, this is SBEINIT.COM, which emulates an SB16 card in software, setting up a 'phantom' card at IRQ 2/5/7/10 which responds to the standard set of DSP commands and ISA bus behaviour.
These drivers would come shipped with your sound card.
*(Not to be confused with a 'DSP' in other usage which provides programmable effects like reverb or echo.)
**(You can read about this in the Sound Blaster Series Hardware Programming Guide)
2) Game support to -consume- the hardware interface
Completely independently of the above, is the 'driver' that provides the game's ability to talk to a specific sound card. This could be more precisely called an audio library, but since it also has to manipulate the Sound Blaster / Windows Sound System etc. DSP processors and motherboard DMA controller on a direct hardware level, it's also a driver. In this respect, a DOS game is sort of like a mini-operating-system into itself.
This driver takes the form of a library of routines that abstract the base primitives of the sound card interface into a useful, consistent set of commands for game developer.
By itself, a Sound Blaster provides a single output stream of audio and FM capability. A Gravis Ultrasound or SB AWE provides a wavetable interface to multiple short looping streams of sound-card-RAM-resident samples (in addition to the SB digitised stream and FM, for the AWE). The PC speaker goes beep.
The game programmer doesn't want to think about this level of detail - they want to start music, play an explosion, etc. It would be the driver's job to abstract these details: begin/stop output, start/stop sound effects, mix them, alter volumes, etc.
Early games would have these drivers coded directly into the game in a sort of ad-hoc fashion - Halloween Harry can only support original Sound Blasters, and the support is hard-coded into the game. Rise of the Triad has its own huge sound library; since RoTT is open source, you can see all of the different initialisation and support routines on Github.
For late, mature MS-DOS games like Theme Hospital, a library like Miles or HMI is used. If you've seen a sound card set up screen with dozens of sound cards available, then they most likely use one of these libraries. I point this out since the different sound card drivers can be shown in a directory listing as
.hmi files. Epic MegaGames Jensen-library games like One Must Fall 2097 and Jazz Jackrabbit store their sound card drivers in