How do I remove this tap?

[seco services]

Theres a few ways to loosen them useally a short sharp wrack on the spanner/adjustable, But be careful you don't put the mixer through the basin.
Can sometimes be more quicker/safer to remove mixer and do it on a rag on floor or a vice.

 

[Madrab]

No you don't remove the 'little brass bit', that's the top of the spindle that runs down through the valve to the washer. Depending on the type of valve, you could have what looks like 2 nuts, smaller one on top (Gland Nut), larger one below it or just below the spindle top there's a circlip then the big nut below. The big nut is the valve body. Grips/large wrench and cloth to hold the tap still whilst using a tight fitting adjustable spanner on the nut and it should eventually move, may need persuasion/WD or worst case as BoB mentioned, heat. That being said, if the taps have got to the point where one drips and no water from the other it doesn't sound like it will be as simple as just changing the washers and it's fixed. There will probably be wear to the tap seat as well, that would need a seat cutter to renew the surface to ensure the new washers seal properly and wld be advisable to strip down the valve body and service it (lubricate and possibly new o rings) to do the job properly and avoid problems in the future.
All in all it may be easier for you to replace the whole tap as scoda has said if you don't want to get the experts in.