The term anime is a broad definition for animation originating inside Japan. When an uneducated person hears the term anime they may think "cartoon" or assume it's for children. This just isn't always the case. Anime deals with much more controversial issues than most of your saturday morning kiddie cartoons. That's not to say some anime isn't for children, there are quite a few titles that would be considered "family entertainment."
The different forms of Anime are often compared to regular; non-animated films and fall under the same genre as they do. Romance and love is a popular one among Anime fans and 'chick flick' goers alike and contain a story based around losing a love and regaining it, falling in love with someone far off or far away and eventually getting them to feel the same, or generally following a story of a growing relationship, as do films in the romance genre that are non-animated. The difference in Anime films however which can often separate them from the non-animated Films is that they can be set in a complete fantasy world due to them being drawn and animated. A world full of dungeons and dragons, war torn lands and huge spanning forests and waters home to many mythological based and imaginative creatures. This is what separates the romance genre in anime films from reality, the fact that every film can be unique but at the same time follow a similar story of pursuing love.
The more popular Action / Adventure genre is also something in the Anime world which can be easily separated from reality action movies. Reality action movies such as Rush Hour follow a kidnapper and drug dealers among other things, whereas an anime version of the film would follow the same kind of story but in more depth and fantasy due to it being drawn out and animated, meaning you can include more high impact action, explosions and many more fantasy actions including the world and it's beasts while following the exact same story in the process. The possibilities of Anime works are endless when the imagination runs wild.
You can draw out a masterpiece of a film with a basic story and it still can be a huge masterpiece due to the amazing effects that can be implemented in Anime films which can't be implemented as easily and in some cases not at all, in non-animated films.
A particular favorite Genre of Anime amongst it's fans is 'Mecha' which uses huge and outrageous, out of the world Robots armed with massive guns and cannons among many other unique weapons and technicalities which you wouldn't see easily in reality films.
A favorite among Children is the likes of Pokemon and Digimon due to the use of 'cute' and unique beasts and monsters, often friendly and child based humor and jokes which easily amuse people of a certain age. Many genres of Anime exist and all of them have their unique fans. So many things can be implemented in anime no matter the genre that can't be implemented in real films, this is the reason that Anime films are so popular and becoming more and more popular all the time. Maybe some day Anime films will even take over the film awards!