This is a complicated topic, and there are many bad answers on this subreddit (and on the internet in general). Trying to explain it by analogy just leads to misunderstandings, so I'm going to break it down into bullet points. I believe each bullet point can be understood on a surface level. Elaboration on any of the points is going to be complicated.
- Our best description of particle physics is the Standard Model. It is based on something called Quantum Field Theory (QFT).
- In QFT, elementary particles are excitations of fields. For example, a photon ("particle of light") is an excitation of the electromagnetic field.
- Mathematically, a field is something that has a value at all points in space.
- Usually a quantum field has a value of zero in a vacuum. The Higgs field is unique in that it has a non-zero value in a vacuum.
- The masses of elementary particles arise from their interaction with the Higgs field. This is a part of something called the Higgs mechanism.
- Higgs bosons are excitations of the Higgs field. They take a lot of energy to create, and they don't live very long. They don't need to be "around" for particles to have mass. The discovery of the Higgs boson was convincing evidence for the Higgs mechanism.