A stay of advertisement is where they withhold all or part of your details in the local papers, gazette and insolvency website.
In order to get the stay of advertisement you have to apply for it at the time of your bankruptcy i.e. when you are at court, you will probably see the judge. I think it cost £30 (it did when I went bankrupt).
You will need an extremely good reason why your details would be withheld i.e. your life would be at risk if details were shown etc. I had a very serious and good reason yet they still printed my name but did put details withheld by the courts. You will also need to provide supporting evidence i.e. police statements etc. etc.