A private class is not simply a yoga class for one student. It's an entity all in itself – with the time to work on goals – spiritual and physical – injury prevention and rehab, dealing with emotional blocks that come out as physical pain or even just squeezing in the time for yourself.
For example – restorative classes in yoga can bring up a lot of emotional "stuff," but after the class we all have places to be and things to do, so there is little to no time to reflect upon how and why the gentle poses in the class, and the breathing techniques practiced have such dramatic effects. This is where private classes come in. Now I know most yoga teachers are not professional therapists or counsellors, but I also know that most yoga teachers come to the practice because it helped with their own healing process.
Private classes can be as asana-focused or meditative as you want or need, and can come with recommendations for home and class practices – modifications, different ways to use props and new breathing techniques. They can be held anywhere (within reason) – most teachers will come to your place or hire a room in a gym or studio. This gives you ultimate privacy with minimal distractions – there's no sneaking a glance at your phone during a private class with the instructor's attention solely on you!
Another reason why one might consider taking a series of private yoga classes is for assistance in deepening and refining their yoga practice. An advanced practitioner may want to work on better alignment or to learn new breathing techniques, they may want to learn the proper use of props or learn how to develop their own home practice. Having someone look at your body in detail, with no distractions from other students can catch any habitual misalignments or release valves that can be missed in a full class of less advanced yogis.
Some people – like myself – love the social aspects of yoga. I practice with my friends on a regular basis, and actively seek out busy classes. I love the energy of unison ujjayi breathing during a sun salutation and catching someone's eye across the room during a wobbly balancing posture. But for other people, it is difficult to focus and relax in a group setting, whether it's anxiety over other people's perceptions or a lack of knowledge of basic postures. For such individuals, private lessons are a great introduction to the practice, building up confidence and also dropping the ego.
For me, losing my ego in my practice was what improved it the most – both physically and mentally. Instead of forcing my body to bend and balance like I'd usually do, I took a class to make "ease" my theme, going to maybe 40% of my capacity and noticing every single detail of the pose, from my feet to my breath. Does this pose make me breath deeper or shallower? Where is the weight in my feet? Where are my hips? Taking a private class with a teacher who is focused on making you feel the best you can possibly feel is such an incredible way to learn how to appreciate your body and all its flaws.