I'll cut to the chase - guitar tuners are an absolute necessity, regardless of your style, experience, or gigging situation. Period. There’s really no more practical piece of gear you can own. While every guitarist should know how to tune his or her guitar by ear (or by another instrument), a tuner is standard equipment for professionals and amateurs alike.
The most common type of tuner is simply a standalone tuner that - at a minimum - has both a 1/4” input for electric guitars as well as a microphone for acoustic guitars. The tuner should be calibrated for “standard” tuning (A440) though it’s not uncommon for many tuners today to accomodate alternate tunings as well as non-standard calibrations. Fortunately, standalone tuners are readily available for under $10 these days. Both Korg and Boss make some really nice tuners, though you can find no-name tuners on eBay and Amazon for next to nothing.
If you’re a gigging or even recording guitarist, you probably want to invest in a pedal tuner. If you have a pedalboard or other pedal effects, it’s easy to incorporate a pedal tuner such as a Boss TU-3 or TC Electronics Polytune (both popular choices) into your pedal rig. The advantages of using a pedal tuner is that you can easily engage a mute while you’re tuning so the audience or your bandmates won’t have to listen to you tune. Most pedal tuners allow you to calibrate to non-standard tunings if so desired. Another good thing about pedal tuners is that many of them (e.g. the Boss) have a built-in buffer, so they will help drive long cable runs.
An increasingly more common guitar tuner is known as a contact or headstock tuner. These are tuners that attach to the guitar body or, more commonly now, to the headstock. They are able to sense the vibrations coming from the guitar and are generally very accurate. Many headstock tuners have low profiles and are hardly noticeable. I use a Snark headstock tuner and swear by it (I have a few of them, to be honest). They are affordable (generally under $20) and make tuning both transparent and easy.
Finally, for both iOS and Android users there are a number of apps available that will use your phone's microphone to sense the sound coming from your acoustic guitar or amplifier. While a tuning app isn't something you're likely to use in a live situation, they can be very convenient while practicing or if you forget your main tuner. I have the TC Electronic Polytune app and find it as reliable as the "real" thing.
Tuners today are so inexpensive that most players can afford to have both a standalone tuner as well as a headstock tuner. There are even free apps available for both smartphones and computers that make tuning a breeze. My favorite tuners are the TC Electronic Polytune and Snark headstock tuner, though I also have a Boss TU-3 as mentioned above. FYI, the Polytune app for iPhone also works amazingly well.