I'll give you what I learned in my apprenticeship. I can't say that I really believe any of this anymore (though I'm still open-minded...), but here it is as I learned it. *cough* This will be a little long...
What I was taught:
1. It is possible to work magick both with tools and without them.
2. However, there are downfalls to both.
3. Though possible and still effective, it is harder to work without tools because it takes a greater mastery over yourself. Thus, unless you are a truly exceptional magician, you will find that magicworking without tools is less effective than with.
4. However, a tool is only as good as the person working with it.
5. In order to work effectively with tools you need two things... two things which, if you master them, will incidentally allow you to also work WITHOUT tools...
6. However, depending on how well you have mastered yourself you STILL may not be "good enough" to be as effective without tools as with.
7. Those two things are Will (which can be triggered by emotion) and visualization.
8. After you have mastered YOURSELF, you can master magic with and without tools.
9. The key to knowing when to NOT use tools is understanding what makes tools generally more effective.
10. There are three reasons why tools are generally more effective: they help you focus your mind, your energy, and they are composed of materials that help with the transfer of energy.
11. I'll illustrate using the example of a wand. Say you want to cast a circle or create some other kind of protective energy barrier. It IS possible to create energy barriers with just your index finger (or, if you are REALLY good at visualization, simply by imagining one rising around you), but if you are still a beginner, a wand helps in three ways:
A) It gets you into the mindset of doing magic and thus make your efforts (and their effects) that much more powerful (I'm convinced that magic, if it really exists, is more than half psychological)...
B) It focuses your energy and helps you channel it (without a tool channeling energy can be tricky... just figuring out where to get energy from and where to send it out of you is hard. Do you draw it from the air and send it out of your fingers? Do you draw it from the ground up through your body and then out of your third eye? Do you just imagine it coming up out of the ground and rising to form a barrier? All potentially work... but some are probably harder than others.) A wand, on the other hand, is "easy"; just decide what you want to do, concentrate on it, visualize, and point.
C) It is composed of certain materials that are conducive to magic work. Take crystals, for example: there are 2 kinds, receptive and projective. Receptive help you to meditate. Projective help amplify and channel energy. Easiest way to tell which one is which (if you are any good at feeling energy) is to hold them in your hands. If you feel like meditating, it is probably receptive. If you can feel its energy sort of tingling along your nerves (or if you point it at your hand and feel the same thing), then it is probably projective. A wand usually has a crystal on the end, so it is no wonder that when you send your own energy and elemental energy through it, the results are probably going to be more powerful than if you simply used your bare hands.
12. However, someone who has really mastered themself (meaning, in this case, really mastered Will and Visualization), can work magic without tools that is probably just as effective, if not more so. This is because the human body is great at channeling energy; you just have to know how to do it. Once you master your Will and Visualization, you can begin learning to channel your own energy until you are as good with your own body as you are with a tool.
13. So, if you are not yet at the level described in #12, when can you ditch the tools and still have strong results? When you are doing small spells, prayer, or meditation (like, if you are sending your well wishes to an ill friend... though a cone of power might work better, which needs to be created in a circle, which would probably be best created with a wand or athame). Also in times of great emotion and times of great need. Those two are often are one and the same. Take Fra.Morelia's bogeyman, for example. If that thing suddenly pops up, Fra will probably be so surprised and shocked that the first thing he (she?) will do IS put a barrier between himself and it (if not outright banish it), without the use of any tools, without any conscious preparation, without moving a muscle. Previous experience and the general adversion that comes from that experience would provide the emotional incentive to do so almost instinctually. (Though too much adversion... like fear, could actually empower the thing... I think Master Vigil already covered that).
So... yeah... that's all I can think of. Sorry if that was a little long!