Horror movies play on one of our most basic emotions: fear. As adults, our fears tend to be rational rather than supernatural, but horror movies take us back to the time when our world wasn't ruled by the rational. This makes horror movies especially good films to analyze.
Watch the movie and notice any recurring elements, such as when people experience the horror. For example, people often wander away from the group in horror movies, only to experience a supernatural fright or an attack. Notice whether the "horror" of the movie is natural or supernatural, human or not, and listen for any explanation of why the horror is happening.
Examine the movie as an example of the horror genre. To do this, research the history of horror films, especially looking at trends in horror movies over the decades and at innovations of individual films. Watch horror movies from across the century and read books like "Projected Fears: Horror Films and American Culture" by Kendall R. Phillips that record elements of the genre's history.
Work with what you can observe as you analyze. The horror genre often uses metaphor and symbol, even when the director or writer didn't necessarily plan it that way. Focus on the repeating elements that seem important to you, or on scenes that you thought were especially good.
Link the ideas of the movie to historical or contemporary events. The horror genre allows filmmakers to comment on these events without being overt about their politics. For example, the classic horror film "Night of the Living Dead" uses the community reaction to zombies to explore race issues and group conformity.
Analyze the sexuality of the movie. Horror movies are often about sexuality, so this topic can provide rich analysis. Think about both metaphor and actual plot as you analyze sexuality. Vampire films are classic examples of sexuality in horror films, as the vampires are often aroused by beautiful women, but their only expression of this desire is to drink the women's blood.
Use Jungian archetypes to analyze the movie by examining how filmmakers tap the collective unconscious to create meaning in the horror genre. In this analysis, horror movies contain the same loaded imagery as do our nightmares. The book "Psychological Reflection on Cinematic Terror" by James Iaccino introduces a full range of Jungian archetypes and how they apply to horror movies.
Apply a feminist analysis to the horror movie by examining the function of gender within the film, as well as the power that women (or females) hold. For example, often women are the victims of the monster of the movie, but in other movies, the woman is herself the monster. Seek out what this might mean about the power that women hold in real life, and what cultural ideas about it might be. You can also look at the way individual scenes are shot as you apply a feminist analysis.
Write down your analysis, even if it's just to discuss with friends later. The best analysis of a horror movie needs details. If you don't write them down, you might forget some of your points.
Analysis isn't the same as opinion. For analysis to work it has to be grounded in concrete detail and you have to be able to logically support your ideas. There are literally hundreds of ways to analyze any horror movie, but these approaches should get you started.