HOG: Hidden Object Game. Games with objects hidden in pictures. The basic HOG has screen after screen of lists of objects for you to find sometimes seperated by dialog and story line or simple puzzles of another type like word games or jigsaws (often refered to as mini-games). 
A good example of an HOG game is the old I-Spy Games such as I Spy: Spooky Mansion
IHOG: Interactive Hidden Object Game. A very popular genre that is part adventure game and part hidden object. Hidden Object Scenes, or HOS, are found by clicking in a particular place in a "location" within the game; a static page which has clickable areas of interest. One or sometimes several of the found objects are useful within the storyline. For example, you may search for twelve items and keep the "flashlight" which then goes into an inventory that you carry with you. Using the flashlight later in another location might give you access to a hidden object scene or progress the storyline by allowing access to the next location. 
Sometimes games are labeled IHOGs if you have to interact with objects in the HOS itself, opening drawers in a desk to search the contents within for instance. There are so many good games in this genre, it's hard to pick just one, so I'll recommend my favorite, the Mystery Trackers series. Left is a link to the first, but there are many more sequels.
FROG: The Fragmented Hidden Object Game is a twist on the IHOG in which you search for parts of an object. Usually there is a story line objective that completes once the object is put together. Occasionally, there are fragmented objects that complete within the HOS as well that allow progress to another part of the HOS. 
These games include walking around and exploring scenes that are linked by story line similar to the IHOGs but instead of carrying flashlights and keys, you are also carrying portions of partially completed objects in your inventory. The Dark Parables Series is an excellent example of a beautifully designed FROG!
TM: Time Management. Just as it sounds, you have to manage tasks within a certain amount of time to accomplish a score or finish a goal. The generic TM games are usually the "Dash" variety (named after one of the first, "Diner Dash"). You often have to click quickly to take care of customers of some sort, such as restaurant patrons, hotel guests or even children in a daycare. There is a wide variety of subtypes but the idea is the same, organize the individual tasks to complete the "day" and win the scenario. Finishing a certain number of days wins you upgrades to take care of things faster or more efficiently. To find the classic TM games, look for keywords like "dash, frenzy, haste, rush, hectic" etc.
One of the newer TM games that I really loved is Cathy's Crafts. In this particular game it's hard to achieve 3 Stars or Gold Time for most levels on a first try. I got as far as I could, then I went back to try to get a better score later. Also, since it's very new, I've linked the Collector's Edition. Cathy's has been out for a while, so hopefully, we'll get the standard edition soon.
Building: Building TM Games focus your tasks on constructing something (a house, a store, or even a toadstool) on an empty lot within a certain amount of time. 
Typically, resource management is part of the game and buildings which produce resources must be balanced with other types to reach the level goals. Royal Envoy is a great introduction to the genre. With a ramp-up into more complex levels, it never gets crazy-hard.
Road: Like Building, "Road" games are a growing subgenre of TM games. Play procedes by clearing roads or paths with "workers" that eminate from a single building on the map to accomplish level goals. 
These goals usually incorporate clearing obstacles from the roads and constructing supply-producing buildings to help you do so. For an easy introduction to road games, try the "My Kingdom for the Princess" series.
TD: Tower Defense is a casual game ported over to normal gaming, or is it the other way around? Either way, these games are a lot of fun. Enemies approach your homebase or HQ on designated paths usually in waves and often at particular speeds per unit. Your defense against the maurauding hoards is a series of towers (normally restricted in placement in casual games) along the roads. The towers fire at the invaders hopefully killing them all before they reach you. Tower type and improvements vary and are the core mechanism behind play as most can be done on the fly during the invasion. 
Another benefit to this genre is that there are more complex types on Steam and other full scale game clients. A casual tower defense game that I really enjoyed is Fort Defenders. It's sea based, so as the invading vessels sail past, the land based towers fire at them attempting to stall and sink them before they reach the coast of the main island.
Match 3: Match 3 games have been around a long time. Remember Bejeweled? The concept is the still the same, on a grid of icons, use some method (switch icons, swap icons or link them) to line-up groups of three or more. Matched groups are then removed from the board and you score points. The goals are various and include clearing a certain number of some type of icon, making matches over areas of the grid or unlocking particular tiles. Power-ups like "bombs" or "hammers" to remove some of the board contents are typically awarded from large matches. 
The level rewards have grown a great deal more complex in the last few years too, often with ways to spent collected coins within the game to decorate gardens, build castles or even make your own aquarium. The "Fishdom" series is one of the best. I've linked the original, but there are more aquariums to be made in Fishdom II and III.
Marble Popper: A vaiety of Match 3 in which a "cannon" fires a colored ball (or other icon) at a moving row of multicolored balls. 
The object is to match three like balls with the one you shoot, causing the match to disappear and the line to get shorter with the end goal of eliminating all the balls in the line within a certain time or before the lines reaches some end point. Zuma is the classic, but I also really like the original Sparkle.
Mosaic? Confusingly, two very different types of game are labeled as mosaic. 
One is a jigsaw puzzle type game that gives you peices to arrange on a board to create a picture. There is usually a timer and points given for not misplacing any of the pieces. Some of these are now called "Patchwork" or "Patchwork Mosaic". Mosaics Galore is a good example.
The other type of "Mosaic" game is logic based and consist of filling in some squares and blotting some out to create a picture. 
The Fantasy Mosaic Series is a lot of fun and there are tons of them.
Other Genres If you look at any game site, there will be more genres than the ones listed here, but most of them are selfexplanatory. Jigsaw puzzles, board games and word games are exactly what you'd expect. Similarly there are no surprises in casino or majong game categories. One note about Card games is that many of the games entitled "Solitaire" are actually a very simple matching type of solitaire (match two like cards) or golf solitaire (remove cards from the board in ascending or descending order). It's rare to see the old Klondike type game included in something labeled "Solitaire".