Partial knee replacements are a good choice if the arthritis is localised to one area of your knee - usually the inner aspect of your knee. Patient satisfaction is generally higher (it feels more like a natural knee) but paradoxically failure (the need for revision surgery) is slightly higher than a full knee replacement. If possible i would prefer a partial knee to a full knee as it preserves more of your own undamaged tissues including ligaments and cartilage. I can tell the difference between my patients who have half versus full knee replacements - partials are usually happier!
Some knee surgeons do not offer partial knee replacements although most specialist knee surgeons do. Take advice from the most experienced knee surgeon. His or Her decision will be based on your symptoms, x rays and examination findings.