Homeopathy has a couple of basic principles.  One is that "like cures like".  For example, if you are suffering from burning, itching, watering eyes, you need to find a substance that causes these same symptoms, because you can cure your symptoms with this substance if it is prepared in a certain way.  The underlying cause of your symptoms doesn't matter because a substance that causes similar symptoms will solve the problem in any case. 

The other basic principle is that less is more.  When toxic substances are diluted to an extreme degree using a specific procedure, they turn into powerful healing agents. The more they are diluted, the more powerful their effect becomes.

The preparation procedure uses a series of dilutions in specific proportions, succussing (shaking) the mixture precisely ten times between each iteration to trigger the desired effect.  Roman numerals (mostly X and C) are used in conjunction with an Arabic numeral to designate the number of times the mixture was diluted and the proportion used at each dilution.  For example, if a preparation is designated as 20X,  it means that the mixture was diluted twenty times with a "one in ten" proportion.  The first dilution would have added 1 part toxic ingredient to 9 parts water, then shaken it ten times. At this point you have a 10% solution with respect to the active ingredient, designated as 1X. One part of this mixture is added to 9 parts water and shaken.  At this point the proportion of the active ingredient to the water is 1 in 100, so you have a 1% solution, designated as 2X.  And so on. The number of X's indicates the number of zeroes in the proportion, and if there's a C instead of an X you should double the number of zeroes, because C indicates a proportion of 1 in 100.  Homeopathy frequently uses solutions of 30X or even higher.  At 30X the ratio of active ingredient to water is 1 in 1,000,000,000,000,000,000,000,000,000,000. That's 30 zeroes folks, a number so big that I don't even know what it's called. My guess is nontillion.

At some point the volume of the preparation becomes so large that it's necessary to start throwing part of it away and perform the dilution with a smaller amount. Some homeopathic remedies require such an extreme dilution that there isn't enough water on Earth to make a full batch. The last molecule of the active ingredient is expected to disappear from the mixture at around the 24X point.  A 30X solution will almost certainly be nothing but pure water, and proponents use the concept of water memory to explain why this would have a healing effect:  the water remembers that it used to be in contact with the active ingredient because its memory was triggered by the shaking between each dilution. Since less is more, the total absence of the active ingredient makes this mixture more powerful than one that contains a few molecules of it.

Vibrational energies are occasionally invoked as a supplemental explanation for homeopathy and other treatment methods that don't have a plausible science-based explanation for how they could work.  These energies are undetectable, but they're presumed to exist because something must be causing the effect and there's not an observable real-world principle that can explain it apart from the placebo effect, which is not an acceptable answer. 

Homeopathy theory runs directly opposite to our science-based observations of how the world appears to work.  There is zero objective evidence for the notion that like cures like, and lots of evidence for the notion that the best way to cure ailments is to find the underlying cause and undertake a treatment program that is tailored to the circumstances.  If you have itching burning eyes and try to cure it with a substance that causes these same symptoms, the most likely outcome is that your eyes will feel worse than they did before. You're more likely to have a good result if you figure out why your eyes are burning (allergies, let's say) and then do something that's been proven to reduce that type of reaction, like avoiding the allergen or taking antihistamines. 

Real world observations indicate that more is more and less is less; you get a more powerful effect when you use more of something, not less of it, and a substance that isn't present can't have any effect at all.  An energy or vibration that is strong enough to cause an effect should be detectable. There is no realistic explanation for how water could possibly have a memory, which is a good thing IMO;  every drop of water on earth has been through a lot of things that we wouldn't want it to remember, for example a flush down the toilet or being drunk by someone who was dying of a terrible disease.  Shaking a container of water isn't known to have any effect on it at all beyond introducing some air bubbles or stirring up any particles that might be in it. This is good too; think of the terrible things that might happen if it actually worked, and a germ or a speck of dust got into the water that was used to make the homeopathic remedy.

Nonetheless there are many people who use homeopathy and believe passionately that is very effective.  The science-based explanation for this is called the placebo effect;  people expect the remedy to help them and so they perceive that it did help them.  The "mind power" involved in the placebo effect can actually be very powerful and can cause a measurable improvement in physical condition.  This is why scientific studies always use a control group taking a placebo when research is conducted on medicines; the simple act of taking something is expected to have a positive effect even if it's completely inert. The real test is whether the medicine performs better than the placebo did, not whether the medicine performs better than doing nothing at all.  The only time that homeopathic remedies have been found to perform better than the placebo was in badly flawed studies conducted by homeopathy proponents who did not follow proper scientific procedure. When proper procedures were followed, homeopathic remedies did not perform better than placebo.

Animals don't have any expectations about what the remedy is supposed to do for them of course, and are more likely to think that you're tormenting them than helping them if you make them take something by force.  But the placebo effect has an influence here too.  The human's perception of the animal's condition will be influenced by the knowledge that the animal took the remedy, and the way the human observes and interacts with the animal may have an influence on the way the animal behaves.

Timing factors can also lead to homeopathy or some other type of treatment getting credit for a cure that it didn't really accomplish.  There are many anecdotes about someone who tried many different methods to cure something, but nothing worked until they tried homeopathy (or whatever). Some treatments take time to work, and the cure might have actually been brought about by something that was done earlier.  The body will often cure itself after an ailment has run its course, and it may have been a simple coincidence that homeopathy was tried at the same time there would have been a natural recovery without any treatment at all. 

Although most homeopathic remedies use extreme dilutions that can't reasonably be expected to do anything, there are some that use a dilution of only 2 or 3 X. That's a concentration of 1% and 0.1%, respectively.  There are some substances that can have an effect at this concentration, and homeopathy always uses dilutions of toxic substances not materials that are believed to be beneficial at their normal strength. I'm not convinced that these relatively high concentrations are harmless; it would depend on what kind of toxin is in the remedy and how much of it is required to produce a toxic effect. If this article hasn't convinced you that homeopathy is useless, then to be on the safe side I would recommend doing some toxicity research before using anything that has a dilution factor of less than maybe 5X (1 in 100,000).

Here are some interesting links:
Science-based Medicine - an article detailing the results of a scientific review that considered more than 1,800 studies, with additional discussion on several other reviews and how homeopathic practitioners respond.