The first type of particles are bosons bosons. Those are these having integer spin. In the standard model, there is the Higgs particle, which has spin zero, and the photons, the W and Z bosons, and the gluons, which all have spin one.
Particles with spin zero are also called scalar particles. Since their spin is zero, the properties of such particles are the simplest when changing to a different coordinate system: They just look the same.
Particles with spin one are also called vector particles. Such vector particles are described like photons. The name vector stems from the fact that under a coordinate transformation the fields describing a vector particle changes in the same way as a line which connects the origin of a coordinate system and an event. The latter line is also called a vector, and hence the name for particles of spin one.
There is actually also a hypothetical particle with spin two, the graviton. Such a particle is also called a tensor particle. Tensors are generalizations of vectors when it comes to coordinate transformations, and fields of spin two particles transform in the same way as such tensors. In general, tensors are rectangular collections of numbers, where the columns transform like a vector under coordinate transformation.
Elementary particles with higher spin are not known. However, particles made up from elementary particles add their spin together (though not necessarily in the sense 1+1=2 - it can also be subtracted, 1-1=0, and everything in between), and can thus have higher spins.
Furthermore, to each such type of bosons, there exists a so-called pseudo bosons, i. e. a pseudo scalar, a pseudo vector (sometimes for historical reasons also called an axial vector), and a pseudo tensor. The difference between a boson and a pseudo boson is what happenes if you reflect the world in a mirror (a parity transformation). Ordinary bosons just become bosons once more. In contrast, the fields of pseudo bosons are multiplied by minus one.
Ok, after all this classification and name stuff, what is special about bosons? The most striking feature is that you can pile them upon each other. That is different from the small balls one often uses to imagine elementary particles: We can stack such balls next to each other, but never ever can two of these balls be at the same place. But bosons can. That is very hard to get in line with our ideas of how things work, and it shows just how quantum bosons are: they behave in a way which is just unexpected.
This is, of course, only true, if the bosons do not repel each other by some force. For example, if you would have two electrically same-name charged bosons, you would have a hard time to bring them together. But if they have oppositely named charges then they would just love to sit at exactly the same place.
In fact, if bosons do not repel each other because of a force acting between them, they have a tendency to lump together - two bosons rather prefer to be at the same place than being apart. This phenomenon is again a pure quantum effect: If you would have two balls, which are not talking to each other, they ignore each other very consequently. The reason for this different behavior is encoded in what physicists call statistics. In case of the boson this statistics is called Bose-Einstein statistics, in contrast to the classical statistics of the balls. Statistics describes how particles distribute themselves. Classical statistics is essentially randomly distributed, but bosons with Bose-Einstein statistics are not entirely randomly distributed but tend to get together.
This property also pertains to a different thing: The energies the particles have. While classical particles have just their energy, independent of every other particle, as long as they do not interact, bosons tend to have the same energy.
The extreme case of getting together is occurring when a sizable fraction of all available bosons are involved, and all of them have the lowest possible energy. That is what is called a Bose-Einstein condensate. This type of stuff is a state of matter similar to being liquid or being solid. But it only occurs under rather extreme conditions, in particular at very low temperatures. On Earth, there is no naturally occurring case of such a condensate. But it was possible to create such condensates in the laboratory using atoms.
In particle physics, such condensates play a central role. The Higgs effect was associated with a condensate of Higgs particles: It is just such a Bose-Einstein condensate. The same applies to the mass generation from the strong force, though in this case it is not the quarks that form a condensate. Since they are fermions, as will be discussed next, this is not directly possible. But states made up from two quarks (or a quark and an anti-quark) can condense. Since spin adds, such states have either spin zero or one, and thus behave like a boson, if one is not looking too closely. And these effective bosons are, loosely speaking, condensing to a Bose-Einstein condensate in this case.
These are only some examples, but such condensates play very often a role, from superconductors to the interiors of neutron stars. Thus bosons, with their strange properties, are very important to physics, and especially particle physics.