I will divide your question into 3 parts
1. Reputation 2. Quality 3. Acceptance Rate
1. Reputation: Apart from Oxford, Cambridge, the LSE, Edinburgh, and Imperial, most people in the USA won’t know the difference between the various UK universities, so it can be either University College London or Warwick or Manchester or Bath all are same reputed. This doesn’t speak to their merits, just awareness. In UK and Europe this university is well known for its quality. Even in my country (Sweden), we know Bath very well.
2. Quality: It's a highly important institution in both it's research and teaching especially in it's science, maths, economics and engineering courses! In terms of its league table ranking, it is currently 4th in the country (according to the Guardian League Table rankings), and last year the Sunday Times League tables ranked it as 3rd). It's entry standards are some of the highest in the country, and it's psychology degree (which I once studied) has requirements which match both Oxford and Cambridge.
3. Acceptance Rate - Bath acceptance rates are 20%. That means every 100 applicants only 20 are successful which makes an applicant harder to achieve this feat. But this is only when Maths and Sciences are concerned. Remaining courses Bath is easy to get into!!!