Alright, let's start from square one. What is a neutrino? Well, it is a neutral particle; in fact, it's a fundamental particle, as far as we know (that is, you can't break it into pieces). Furthermore, it's a fermion, meaning it has half-integer spin (specifically, 1/2) and obeys the Paul exclusion principle. The neutrino is a particle of extremely low mass, so even with a relatively small momentum gain, it will be accelerated to relativistic velocities. Finally, it's a lepton and comes in three flavors associated with the electron, muon, and tau particle (I'll try to explain this in a bit).
Why does it interact so weakly as compared to other particles? Consider the fundamental forces. It has no charge, so it can't interact electromagnetically. It's not a hadron, so it can't interact via the strong force. It's of extremely low mass, so it can't decay into anything else. Other than gravity, all that remains is interaction via the weak force which, true to its name, is extremely weak at low energies.
Now, about the oscillations. This connects to what I was saying earlier about the different flavors of neutrino. Neutrino flavors are each "connected" to another lepton, each fundamental spin-1/2 particles with negative charge. The least massive (and most stable) of these particles is the electron, so there is an "electron-type" neutrino. There is a quantum number, called lepton number, which basically counts the number of leptons of a certain type and which is approximately conserved in all interactions. As such, when neutrinos come out as the product of an interaction, they come out as eigenstates of flavor, meaning they can definitely be associated with one lepton type.
For most particles, the eigenstates of flavor and mass are the same, meaning we can measure both quantities simultaneously (that is, we can say that an electron is an electron-type lepton and has mass 511 keV). However, with neutrinos, this is not the case, and the eigenstates of mass will actually be composed of some linear combination of the eigenstates of flavor. These mass eigenstates are the ones you see numbered as "1", "2", and "3". Thus, when you make a mass measurement, the neutrino wave function collapses into one of these eigenstates of mass. If we were to immediately follow it with a measurement of flavor, then there would be a non-negligible probability of obtaining each of the three flavors of neutrino. This is effectively the same as the problem you run into when you try to measure both the position and momentum simultaneously.
Neutrinos coming from the sun start their lives as eigenstates of flavor (specifically, electron-type) and evolve as they pass through the sun's envelope. We originally thought that we would be measuring only electron-type neutrinos from the sun, so our experiments were only designed for detecting them. When our initial measurements came up short of the expected number of neutrinos (as calculated from solar models), we became puzzled. As more precise measurements were made, we discovered that the difference could be made up by including measurements of all of the other types of neutrinos. Since we still think that the neutrinos started their lives as electron-type, the detection of the other types means that there must be some mixing of the flavor states. In order for this to occur, neutrinos must have mass.