Coronal Mass Ejection is a large ejection of mass from the sun. These ejections are so large that they increase the particle density of the solar wind and can be very dangerous to spacecraft. The size of a CME can be much larger than the sun itself.
Solar flare are like CMEs except smaller, more localized, and unlikely to eject enough mass to do harm. They are little flare ups on the surface of the sun that are much smaller than the sun itself.
Here is a webpage all about this big difference.
Solar Particle Events are events in which particles, mainly protons, are accelerated due to the intense magnetic activity due to a flare, or possibly due to a CME interacting with planetary/solar fields. SPEs create streams of protons that are so energetic that they can penetrate Earth's magnetic field. Generally speaking, an SPE can cause interesting things, like an increase in high energy proton radiation in the upper atmosphere/low earth orbit, and therefore can be a safety concern. More information here. One of the bigger problems with SPEs is an increase in RF radiation, which causes some disruption to radio communications over very large distances.
Solar storm is a misnomer and unfortunately I'm not enough of a space weather scientist to fully understand it (in the sense that I'm not a space weather scientist). However, from what I gather, they are actually geomagnetic storms - that is, a disturbance of Earth's magnetic field due to solar activity. They can be caused by large CMEs, but also by other events that involve either the solar magnetic field or high energy charged particles.
Geomagnetic storms are dangerous and we are concerned about a big one happening. These can be responsible for destroying unprotected electronics, as was seen in the 1859 telegraph disruption and the 1989 power grid disruption. However, these events are due to the Earth's magnetic field, not the sun per se. Instead, the sun interacts with the magnetic field causing it to fluctuate, inducing undesired currents.
