These are stems cut from one of the Leucadendron laureolum hybrids; they are sold by florists usually in autumn/early winter in the northern hemisphere. The plants are shrubs in the Protea family that grow in warmer parts of the world - they do not tolerate much cold, some detail here https://www.burncoose.co.uk/site/plants.cfm?pl_id=5862. There are many shades of foliage with these plants, but the one you usually buy as a cut stem is similar to 'Burgundy Sunset'.
As cut stems, they will last in water for quite some time - I've had them for up to 8 weeks, because they tend to dry out rather than flop and shrivel. Just change the water every 2 or 3 days, as you would with any bunch of cut flowers. They are usually displayed with other flowers for contrast, but Leucadendron stems will outlast ordinary cut flowers such as roses or chrysanthemums.