There are basically two ways to go.
First is a special adapter headset, there are many from different manufacturers available, for example Neco H834.
The other way is a reduction sleeve, typically made of aluminium, that you press to your 1-1/8 tube and it makes it 1" headset ready.
Each way has its own pros and cons. Essentially using the second way you are able to run high-end headset, since they don't really produce high-end adapter sets, but there are plenty of racing bike 1" sets out there. But for a regular purpose using an adapter is easier and slightly cheaper, unless you already have a 1" headset you want to keep.
All mentioned above is true for modern standard EC30 and EC34 headsets. If we are talking about older American standard headsets (EC33, something present on vintage cruisers and 90's BMX) then they won't fit to a regular adapter, there has to be a special custom machined adapter for them.
If you need help sourcing any of the parts mentioned above - feel free to let me know, i can help you get them