Because, to my absolute surprise, no one has asked this question before, I would like to elaborate a bit and summarize the most well-known versions* of idealism, as well as popular protagonists of different views, for reference to other questions that may come.
*David Chalmers, in one of his famous articles "Idealism and the mind-body problem", divides the versions of idealism differently, using a micro-macro-cosmic definitions. I will not address this set of definitions here, but it should be remarked that he managed to put most of the idealists under his definitions so it's worth checking out.
Note: unless stated otherwise, all quotations are from Wikipedia. I have chosen Wikipedia instead of superior websites like SEP simply because Wikipedia summarize very efficiently, and what I'm presenting here is a general summary, and not an elaborated explanation.
First of all, let's start with what is idealism.
In philosophy, idealism is the group of metaphysical philosophies which assert that reality, or reality as humans can know it, is fundamentally mental, mentally constructed, or otherwise immaterial.
Basically idealism is the position that says the world, reality, has mental properties**. What is "mental" in this sense? That's a bit vague definition, which is one of the problem of idealism. Commonly when we use mental we mean spiritual, but the best definition that can be used for idealism is "immaterial", meaning that Idealists claim there is "more" to the world than what we can physically explore. This position is famously contrasted against the common-sense materialism.
But what do we think is the limit of "mentality", or "ideas", where can we find it, and how it operates? That's the main bone of contention in the idealism tradition.
Subjective idealism, or empirical idealism, is the monistic metaphysical doctrine that only minds and mental contents exist.
The notorious view that claims that everything that exists is mental. The popular claim of this view (Berkeley's, "Esse est Percipi", to be is to be perceived) is that everything depends on what we perceive, i.e. all that exists is what we perceive. A more extreme take on this position turns to solipsism and claims that only I exist.
It is indeed an opinion strangely prevailing amongst men, that houses, mountains, rivers, and in a word all sensible objects have an existence natural or real, distinct from their being perceived by the understanding. But with how great an assurance and acquiescence soever this principle may be entertained in the world; yet whoever shall find in his heart to call it in question, may, if I mistake not, perceive it to involve a manifest contradiction. For what are the forementioned objects but the things we perceive by sense, and what do we perceive besides our own ideas or sensations; and is it not plainly repugnant that any one of these or any combination of them should exist unperceived?
(George Berkeley, A Treatise Concerning the Principles of Human Knowledge, 1710)
Proponents of subjective idealism (at least to some extent): George Berkley, Hume, early Fichte.
Objective idealism is an idealistic metaphysics that postulates that there is in an important sense only one perceiver, and that this perceiver is one with that which is perceived.
Not the best definition by Wikipedia. Objective idealism, in contrast to subjective idealism, claims that albeit fundamentally having mental properties, the world is real and independent of the subjective perception. Many misjudge objective idealism to be similar to realism, but the only necessary connection between the two is the admission to the objectivity of reality; fundamentally the two disagree about the basic elements that make up our world.
Because this position has very general requirements, there are quite a few philosophers that align themselves with it, but they still disagree on many points, like what is exactly "mental"/"idea", is it a substance? Is it a property? How come does the objective reality connects to the subjective one? Etc. Those discussion brought different interpretations to objective idealism.
Objective idealism … interprets the spiritual as a reality existing outside and independent of human consciousness.
(Oizerman, The Main Trends in Philosophy, 1988)
Proponents of objective idealism: Schelling, early Hegel, Pierce, Royce.
Absolute idealism is an ontologically monistic philosophy, [an account] of how being is ultimately comprehensible as an all-inclusive whole (das Absolute).
Absolute idealism is, in some way, an attempt to answer one of the major questions of objective idealism - how is the connection between the subject and the objective world is possible? The answer, most famously associated with Hegel, is that the world itself is ultimately One, single encompassing being (whether it be an actual being, a spiritual one, or a conceptual one, is a debate amongst the proponents of this tradition).
As stated, this view is commonly associated with Hegel, the most famous proponent of it. This sparked the famous Neo-Hegelanian tradition and British Idealism; which in turn brought life to analytic philosophy as backlash from the Hegelians' teachings.
G.E. Moore took the lead in the rebellion, and I followed, with a sense of emancipation. [Absolutism] argued that everything common sense believes in is mere appearance. We reverted to the opposite extreme, and thought that everything is real that common sense, uninfluenced by philosophy or theology, supposes real.
(Bertrand Russell; as quoted in Klemke 2000)
Proponents of absolute idealism: Hegel, Schelling, Green, Bradley, Wallace, Royce.
Transcendental idealism... maintains that human experience of things is similar to the way they appear to us—implying a fundamentally subject-based component, rather than being an activity that directly (and therefore without any obvious causal link) comprehends the things as they are in themselves.
Kant's idealism is an attempt to overcome subjective idealism (particularly Descartes, Hume, and Berkeley). In order to do so, Kant suggests that while the world as we see it is indeed subjective, meaning what we perceive is completely our own perception of the world (phenomenon), the World itself, the objective world, still exists, but we simply cannot perceive it (it is "a thing in itself", noumenon). This way Kant holds both ends of the line, keeping the reasoning of subjective idealism while maintaining the reality of the objective world.
Transcendental is the philosophy that makes us aware of the fact that the first and essential laws of this world that are presented to us are rooted in our brain and are therefore known a priori. It is called transcendental because it goes beyond the whole given phantasmagoria to the origin thereof.
(Schopenhauer, Parerga and Paralipomena, 1851)
Proponents of transcendental idealism: Kant, Hermann Cohen, Cassirer, Hartmann, and truly many other Neo-Kantians.