Delta One refers to the way a bank hedges its long and short exposures across a portfolio of investments.
Delta One desks focus on 'delta hedging', a very common investment-banking activity. Usually it refers to the way a bank hedges its long and short exposures across a portfolio of investments that may include assets such as shares, as well as derivatives such as futures and options.
To make a portfolio price neutral', you need an overall delta' of zero. That means for every 5% your long positions gain (say stocks you own), your short positions (created using, for instance, short futures or put options) also lose 5% and vice versa, so your overall gain or loss is zero.
In reality, perfect delta hedges are hard to achieve and maintain. In the UBS case, some trades may not have been delta hedged at all.