The term "Agrahara" in ancient and early medieval India referred to tax-free villages that were possessed by Brahmanas. This means that these villages, which were typically agricultural in nature, were exempt from paying taxes.

Option 1, which suggests that Agrahara denoted Jaina monastic establishments, is incorrect. Jaina monastic establishments were known as "matha" or "math" in ancient India, and Agrahara did not refer to these establishments.

Option 2, which suggests that Agrahara denoted secular land grants, is also incorrect. While land grants were a common practice in ancient India, Agrahara specifically referred to tax-free villages possessed by Brahmanas, rather than secular land grants in general.

Option 3, which suggests that Agrahara referred to tax-free land gifted to temples, is also incorrect. While temples did receive land grants in ancient India, the term Agrahara did not specifically refer to land gifted to temples, but rather to tax-free villages possessed by Brahmanas.

Therefore, the correct answer is option 4, as it accurately describes what Agrahara denoted in ancient and early medieval India.