In Harry Potter's early years, when he's taught the 'Wingardium Leviosa' spell, the spell require some wand movement. If you get the movement wrong, the spell will not work or it will malfunction. But in some cases there's no sign of wand movement: they just point-and-say the spell to cast it. This is especially true in the films. Does every spell require wand movements?
When Harry reads "Levicorpus" from the Prince's book, there's no mention of wand movement instructions. He just points the wand at Ron and bang, it works like a charm (or curse...)
@KeithS has mentioned that Harry does in fact make an upward movement when using Levicorpus. So instead I'm going to quote other spells:
From The Goblet of Fire, when Moody is showing the unforgivable curses to the students:
Moody pointed his wand and whispered:
Pointing back at the spider, whispered:
Moody lifted his wand [...]
This is the first time that the spells are explicitly introduced in the series (Avada Kedavra being a somewhat exception), but the book is not describing any kind of wand movements from Moody. He just points his wand at the spider (not even a flick is given).
So no, not all spells seem to require wand movements.
As a side note, I'm under the impression that future uses of Wingardium Leviosa in the books don't describe any wand movements from the users explicitly. This might be just to simplify the reading though.
The basic implication throughout the series, expanded on in Book 7, is that the wand itself is a semi-sentient thing that interprets the will of its master witch or wizard. The spoken words and physical gestures are needed in the beginning to help the wand understand your intent, and also to add power to a spell.
However, once the wand comes to know and understand its owner, and the owner better understands magic's capabilities (and limitations), the wand is able to sense the intention of the owner and channel his or her magical energies to perform the desired effect with little or any other urging. It is implied that nonverbal spells, wherever they differ from somatic ones, are overall less powerful. For instance, in Book 5, one of the Death Eaters in the Ministry is silenced but still stuns and critically injures Hermione with a spell; if the Death Eater had been able to add the vocal component, it probably would have killed her. It would thus be assumed that a spell cast by pointing and saying, or pointing and thinking, would similarly be less powerful than a spell cast with the appropriate gesture.
It's also shown that many wand gestures are very simple; a slash, a flick, a pointed jab, etc, meaning that, as shown in the movies or described in the books, the actual wrist movement might be lost in the character's overall movement.
I believe that the wand as well as the saying of the spell was always just a way to channel your magic into doing what you want. When Harry went to the zoo with Dudley he made the glass in the reptile exhibit disappear then reappear, all without wands or words. He was emotional and was able to channel his magic through his emotions.
In later years at school, witches and wizards need wands and incantations to help channel their magic into learning these new and more complicated spells. Harry learns later that there are silent spells that you can cast just by thinking the incantation. This shows that the words themselves are seemingly unnecessary, you only need to focus on what you want the magic to do.
I would venture to guess that the more magical ability and focus you have, one could cast any spell without even a wand. Unfortunately I can't remember any evidence of this from the books.
Personally, I'm inclined to believe that most of the students at Hogwarts hadn't done any intentional magic prior to their first year and sorting. Therefore, they are taught the wand movements, such as "Wingardium Leviosa" swish and flick, in order to bond the wand to the student. They all had just gotten their wands maybe a few weeks before departing to Hogwarts. The wands are new to them, and they are new to the wands, thus neither are attached strongly to each other. I think the movements become somewhat less relevant later in the series, because it is in fact sentient, and it can begin to understand its owner's intentions by body language or verbal conversational language, therefore nullifying the wand movements. The wand movements are needed, but only as the witch or wizard is learning and growing in their magical abilities. Hope this helped, and correct me if I'm wrong about anything.
It should be noted that apparition is a spell. No mention of want flicking, or use in general is required for apparition (IIRC Hermione apparates several times in last installment without any mention of a wand).
Additionally, the transfiguration used in becoming an animagus is a spell, and you can bet that Sirius was not allowed a wand of any sort in Azkaban, so unless he fashioned one out of something from a Dementor-part (which I suspect would be formidable for dark magic?), then he didn't need a wand either.
I would say the wand is kind of like a dog. When you teach your dog a trick (learn your wand a spell), you first have to learn it to him by using your voice and making a gesture or movement. Later on, your dog knows the trick just by you saying the words. This is almost the same with your wand, I think.