The visa application has to be accompanied by a fee of $131. There's an additional fee for the courier service to return a passport. An appointment has to be booked at the US embassy in London. (The current minimum waiting time for appointments is around 3 weeks but some people have had to wait several months just to get an appointment). The drugs convictions will automatically bar your mother and boyfriend from entering the USA but they can apply for an exemption. Processing of the exemption application takes a minimum of three months, but often far longer.
In practice, your mother has only a slim chance of getting a visa. Your boyfriend has no chance whatsoever. (Someone has posted, here on AB, to say that they were refused a visa for two convictions of driving without insurance. The US authorities are likely to regard drugs convictions far more seriously).
Applying for visas is pointless, because they won't be issued. But, if you want the links, they're here:
http://www.usembassy.org.uk/cons_new/visa/niv/ apply.html
and
http://www.usembassy.org.uk/cons_new/visa/niv/ add_crime.html
Chris