There are some slight differences, so let's let the dictionary do the explaining for us:
So what exactly are we looking at? There are a few things to notice here:
- 招待 means to invite someone somewhere as a guest, or to have someone come from somewhere for some special purpose. Note that this is both a noun and a verb, so it can also mean invitation.
- 招く is very similar to the verb form of 招待, as shown in its second definition. However, it has some other meanings. snailplane mentioned in the comments that she associates 招く with 招き猫. These are little creatures that sit outside a shop and beckon guests and customers in with their waving gesture. This is also 招く, as shown in the first definition, where you beckon someone with some kind of signal. The third sense is just a polite way of saying to have someone come, but the 4th also shows 招く being used in the metaphorical sense of inviting disaster or other such misfortune, similar to how we use it in English.
- 誘う, as also mentioned in the comments, is about getting someone to do something with you, as in the first definition. The second definition is similar to the last of 招く, about soliciting tears or being lured by good weather or whatever onto a vacation. The last is the sense of being tempted to do something that you shouldn't otherwise be doing.
I basically paraphrased the definitions in English, but that should be a clear explanation of the differences.