It is Plectranthus scutellarioides or coleus .
I don't have one, myself, so I cannot give any info about how to keep them.
Checking Google, like you suggest seems a very good option!
gardening.stackexchange.com/users/19430/benn
I've just realised where you are....
this plant is a tender annual, usually grown in the summer outdoors as bedding in the UK.
It can be kept as a houseplant, but needs pinching out regularly to keep it bushy.
Yours needs some attention; first, has the pot it's in got a drainage hole?
If not, it needs to be in something with a hole in the bottom to allow water to drain way.
Second, I recommend you cut it back by two thirds/three quarters, down to about 6 inches or where there are side shoots.
This is best done in a month or two, as spring begins/approaches, but can be done now.
Give it something like Baby Bio houseplant food about once every two weeks after you've cut it back.
This should encourage new shoots from the roots - if you see little flowers forming, pinch those out between your thumb and finger.
The flowers are relatively insignificant and this plant is grown for its highly decorative leaves, so removing the flowers encourages more leaves.
Likes some sunlight, otherwise very bright daylight, and the usual rules re watering apply - water when the surface of the potting soil is just dry to the touch, water thoroughly and tip away any excess in an outer pot or tray after 30 minutes.
gardening.stackexchange.com/users/1380/Bamboo
It grows very well in moist, shaded areas in the outdoors in Indiana during the summer months.
My mother left a pot of it on the brick paved patio and it quickly spread by seed over a large area.
I've tried growing it indoors but didn't provide enough sun.
The colors other than green didn't show as well.
gardening.stackexchange.com/users/20579/Chris
Bamboo's right -- pinch back a lot to get the leaves to show.
consider buying several kinds and putting them together in a window box-shaped place; they can take a bit of shade and I used them as houseplants for years till i discovered that where summers are warm they do well outside as annuals.
(They're houseplants in Seattle, which has the same climate as much of Britain, and outdoor plants in Iowa, which has the same climate as hell.)
They look especially fine with sweet potato vines and lime green foliage of various kinds.
They like water fairly regularly, though they're not swamp plants.
Yours is far too leggy; I didn't recognize it at first.
Cut 2/3 off, then cut each of the long pieces into 4" pieces.
Take off the bottom-
most pairs of leaves (about 1/2" worth) and stick in a glass full of water which covers the denuded parts.
Every couple days, check on the water to be sure it's still covering that part; if it starts getting yucky looking, replace with fresh water.
After a bit (weeks, not months)
roots should be growing on each of those parts.
You will end up with a TON of coleus plants to do whatever you wish with -- build a forest of houseplants or a garden of bedding in the sun.
gardening.stackexchange.com/users/20599/Kes Sparhawk
I have one, although didn't know the name.
Can provide a little of my experience with it.
Apologies that i'm no expert.
It likes a bit of water ever now and again, it can cope with being quite neglected and water will bring it back again.
It is incredibly easy to take cuttings from - literally cut a piece off and put it in another pot with some water and it will probably take.
Every time i've tried to plant one outside (UK)
it has quickly died, I have a feeling its maybe fussy about PH levels.
gardening.stackexchange.com/users/20030/Tim
What is this plant that I was given, please?  If anyone could provide a particularly good guide to caring for it, that would also be great, but I'm expecting to just Google its name once I know what it is :)
What is this purple-leafed (potted) plant?
gardening.stackexchange.com
<identification><houseplants>
gardening.stackexchange.com/questions/36768
gardening.stackexchange.com/users/17026/Brondahl
It is a Plectranthus scutellarioides. It is an annual usually grown outdoors.   It is  Plectranthus scutellarioides  or  coleus.This is a tender annual, usually grown in the summer outdoors as bedding in the UK.
 It is  Plectranthus scutellarioides  or  coleus . This is a tender annual, usually grown in the summer outdoors as bedding in the UK.