A leaking valve
First, drain the radiator in question until the water level is below that of the leak. Turn off the lockshield and the supply valves, using towels to catch any water that dribbles out in the process.
Using a spanner, undo the union nut, then open the bleed valve to release any trapped water. Using PFTE tape (and plenty of it) wrap up the valve tip before tightening the union nut and reopening the lockshield and bleed valves. Once your radiator has filled again, you can close the bleed valve.
A leaking radiator spindle
A far easier job, a leaking raditor spindle can often be fixed simply by tightening the gland nut with a suitably sized spanner. If that fails to work, you can always undo the nut instead and wrap a bit of PFTE tape around the spindle to make a tighter seal.
A leaking radiator pipe joint
Much like leaking spindles, leaking pipe joints are usually a matter of tightening the joint with a spanner. Again, PFTE tape can be used to pad out the join for a tighter fit, if this first option fails.
Many of these fixes may end up being little more than a stopgap. As with most things in life, as a radiator gets older it begins to deteriorate and those leaks get more frequent and more serious. While the fixes listed above might buy you a few more months, now might be the time to invest in a brand new radiator.
At UK Radiators, we carry a wide range of designer radiators that you find in our online store, ready for fast delivery across the UK. Indeed, most orders placed before 13:00 can reach you by the next working day.