I have an indoor plant (sorry, no ideas what type), and it's starting to look sickly. I'm hoping for advice. It's in a relatively dark room, but gets moved to the window for sun every so often.
Your plant appears to be a Fatsia japonica, and it has acquired a scale infestation. Scale can be impervious to insecticides, but there other ways to treat the condition. Yours looks as if there are quite a lot of scale,so spraying might be the best way forward - for a light infestation, it's easiest to try simply wiping them off, and for those which are firmly adhered to the plant, to dip a cotton bud in alcohol,touch the scale and lift it off, preferably without getting the alcohol on the plant tissue. You can make the spray by mixing insecticidal or gentle/biofriendly unperfumed household liquid soap and vegetable oil, more information on that, and other options here http://pestcontroloptions.com/insect-control/get-rid-of-scale. It has general information first, so you'll need to scroll down to the houseplant section.
Fatsia japonica is an evergreen shrub hardy outdoors down to -15 deg C, with an ultimate height and spread up to 4m - its fairly rapid growing, and usually only makes a relatively temporary indoor plant. Yours looks like it will need a larger pot perhaps by next spring; as long as you can keep providing a larger pot ongoing, it will be fine indoors, but will inevitably get to the point where it starts to suffer from having its roots contained over time. It prefers partial shade or shade conditions, so keep it out of direct sun in the middle of the day during summer, even indoors.