So you've lost a loved one recently, and everything for the past couple weeks has been kind of a blur, and you're only just now cluing in to the fact that you've got a plant in your home that wasn't there before, and you're wondering what you're supposed to be doing to take care of it.
We'll get to that, but first let me just say I am genuinely sorry. Not that I know you, not that I knew them, but still. Everybody pretty much agrees that death is confusing and painful and hard. The universe is often appallingly stupid that way. I can't do much about this except sympathize, which I do, and help you deal with the plant, which I will. So.
Often, florists place plants in plastic-lined baskets, or wrap the pot in colored foil. The purpose is to make the plant more attractive by hiding the plain plastic pot the plants are grown in, and catch any water that drains out of the plant when it is watered, so it doesn't wet carpet or tabletops or whatever. However, this is a very bad thing for the long-term health of the plant. If your plant is in foil, remove the foil and throw it away. Plastic plant saucers are available to protect carpet or furniture, if you need them, or you can use a plate or a plastic lid or whatever. (I use plant saucers at home.)
If your plant is in a plastic-lined basket, you can either remove the basket (my preference) or place something (like a brick) in the bottom of the basket to raise the pot up, so that when you water the plant, it won't have to stand in the water that drains out. The latter is probably a temporary solution: usually the plastic is pretty flimsy, and tears sooner or later, and then the water leaks out anyway.
The reason this is important is, although all plants need water, the roots also need oxygen. The soil of a plant that stands in water will be waterlogged, which prevents air from reaching the roots. If the roots die because they're too wet, then they can't take up water for the rest of the plant, which means that the rest of the plant also dies, sooner or later. (Worse: when the roots die back and the plant is unable to take up more water, the plant often wilts, which a lot of people think means that the plant is too dry, so then they dump more water in.)
Gift plants may also be planted in decorative pots lacking drainage holes; if you receive such a plant, you need to repot it into a pot that can drain. (There are plenty of attractive options with drainage holes out there.) It's technically possible to grow plants successfully in pots that lack drainage holes, but it's much harder to know how much water you're adding and how much is there already, so it's very easy to overwater and end up with a waterlogged plant.
For anything after this, it becomes important to know what plant(s) you have. Florists tend to use the same things over and over as condolence plants, leaning heavily on those of a certain size, usually with rounded leaves (sharp, pointy, jagged shapes are apparently considered unsettling), and a shape that isn't too unusual or dramatic. Normally, it's a single kind of plant, but occasionally dish gardens, with multiple plant species potted together, may be sent. Since there's such heavy reliance on the same plants over and over again, there's a pretty good chance that I can guess what you have without having to see it. Here are the usual suspects.
Peace lilies (Spathiphyllum cvv.) are the king of condolence plants, I suspect mainly because of their common name. Plants vary in size and color but always have white flowers, held above the tops of the leaves. Read about how to take care of peace lilies at the Spathiphyllum cvv. profile.
The other very common condolence plant in my personal experience is the rubber tree, Ficus elastica, particularly the variety 'Burgundy,' which is actually nearly black. Read about how to take care of rubber trees at the Ficus elastica profile.
Chinese evergreens (Aglaonema cvv.) have fairly large leaves, patterned in varying shades of gray, silver, and green. There are many varieties, which differ in size, color, or patterning. A few new varieties are being introduced which have pink, red, orange, or yellow in the leaves as well, though the green/gray varieties are still much more common. Read about how to take care of Chinese evergreens at the Aglaonema cvv. profile.
Dumb canes, or dieffenbachias (Dieffenbachia spp.) are similar to Chinese evergreens, but tend to be white or yellow where a Chinese evergreen would be gray or silver. They are also poisonous, and are not a good choice if you have pets or children who might chew on the leaves by accident. Read about how to take care of dieffenbachias at Part II of the Dieffenbachia cvv. profile, and learn about their toxicity and history in Part I.
Crotons (Codiaeum variegatum) are very diverse. Leaves can be nearly any color (red, orange, yellow, green, dark purple, white) and shape ("leaf-shaped," long and skinny, nearly round, threadlike), with spots and streaks all over. They can be kept as long-term houseplants, but they're more demanding than the others on the list. Read about how to take care of crotons at the Codiaeum variegatum profile.
Pothos (Epipremnum aureum) is an atypical funeral plant; florists usually stay away from trailing plants for these situations, because of the possibility that the vines will be stepped on. One sometimes sees specimens that have been trained to climb a post in the center of the pot, though, and it's a common addition to group plantings. Pothos is a very robust indoor plant that does well for most people. Several varieties are available, with white, gray, or yellow variegation. Read about how to take care of pothos at the Epipremnum aureum profile.
Small specimens of the above plants are also frequently used in mixed containers. Some other common small plants for dish gardens are below.
English ivy (Hedera helix) is a trailing plant that I haven't written about much yet. I don't have a lot of experience with this plant, because what experience I have is pretty uniformly bad. Does best in a cool, humid spot with bright light but no direct sun. They're sensitive to dry soil, so don't let them get overly dry. English ivy is also very prone to spider mite infestations.
Heart-leaf philodendron (Philodendron hederaceum) is another trailer. It looks a lot like pothos, but they really are different. I have written a post about how to tell the difference between heart-leaf philodendron and pothos here, and the profile for Philodendron hederaceum is here.
Parlor palm (Chamaedorea elegans) is another plant I haven't written about very much, due to bad experiences with them. Care is basically the same as for English ivy, but unlike English ivy, they prefer warmer temperatures.
Arrowhead vine (Syngonium podophyllum) will eventually trail, but the plants used in dish gardens are usually small juvenile plants, which have an upright habit. The Syngonium podophyllum profile is here.
Baby rubber plant (Peperomia obtusifolia) is a plant I actually like, but haven't written about yet. They do best with some direct sun, but will survive fine in bright indirect light (the variegation may not be as intense, though). Let them get fairly dry between thorough waterings. Plants will suffer if exposed to extreme temperatures (above 90F/32C or below 60F/16C). Despite the name, they are a different species from rubber trees, and aren't actually "baby" anything: that's a full-grown, if young, plant in the picture.
Madagascar dragon tree (Dracaena marginata) is a very common dish garden plant, which I talk about in its profile.
Ribbon dracaena (Dracaena sanderiana) is the same species as "lucky bamboo," but the variety used in dish gardens is almost always variegated. I find them hard to kill, and also hard to make interesting or attractive, as a separate plant. The profile for Dracaena sanderiana is here.
Nerve plant (Fittonia albivenis) is a small, attractive plant with veins that can be white, pink, or red. They shouldn't be allowed to dry out (they'll wilt dramatically, but will revive if watered -- most of the time), and prefer high humidity. Some sun is okay, but they're happy with bright indirect light too. Temperatures should always be at or above 60F/16C. Fittonias are easiest to grow in terrariums. The profile for Fittonia albivenis is here.
Pineapple dracaena (Dracaena deremensis 'Janet Craig Compacta'), like all of the Dracaena deremensis varieties, is a tough plant that handles neglect well. Though 'Janet Craig' is a different, larger variety, most of what is said in the Dracaena deremensis 'Janet Craig' profile also applies to 'Janet Craig Compacta.'
Friendship plant (Pilea involucrata) is much like Fittonia -- it would be happier in a terrarium, though if you water when the soil is moderately dry and give it decent light (bright, but not necessarily sun), and keep the temperature above 60F/16C, it's fairly easy to grow.
Prayer plant (Maranta leuconeura cvv.) is so named because the leaves change position during the day: they're horizontal during the day and vertical at night. See the Maranta leuconeura erythroneura profile here.
Polka-dot plant (Hypoestes phyllostachya) is a fast-growing, weedy plant with leaves covered in white, pink, or red polka dots, hence the name. It needs a lot of maintenance but isn't hard to keep alive. The Hypoestes phyllostachya profile is here.
Umbrella tree refers to two different species in the genus Schefflera, but only S. arboricola is a common dish garden plant. Some varieties have white or yellow variegation. They do best when allowed to get almost completely dry between thorough waterings, and should be kept above 50F/10C. Scheffleras are prone to spider mites, particularly if grown in direct sun in a room with dry air; they don't mind bright indirect light, though.
If you have received a plant you are unable to identify from this list, I'll do my best to identify it and direct you to good care information if you can e-mail me a photo at firstname.lastname@example.org (remove the three 7s from the address first; they're there to foil spambots).
You may also wish to read:
Funeral Plants, part I (for the senders)
The Ten Houseplant Commandments: covers most of the basic mistakes people make with houseplants.
Repotting Questions (With Answers!) Part I and II: covers how, when, and why to repot.
How Often Should I Water This?: about watering plants on a strict schedule, and why you shouldn't do it.
Caring For Unknown Tropical Plants: what to do with houseplants you can't identify.