To lobby is to seek to influence legislators about a certain cause. 

According to Oxford Languages, it originated in the mid 16th century from medieval Latin (lobia, lobium) a ‘covered walk, portico’. It seems to have been derived from the practice of people who frequented the lobby of a house of legislature, using the space to talk with legislators about supporting a certain cause before they went into the chambers.

Which brings us to a further bit of definition. Lobbying refers to trying to convince legislators or other elected officials about a specific issue or matter. For example, if I contact my local representative because I’d like them to vote “yes” on a specific bill (say, that will increase the minimum wage), I am lobbying. If I write to them and express the need for better wages and explain how hard it is to live on a minimum wage job, I am advocating. Advocating takes the form of education on particular subjects.

Organizing or attending an event which rallies around the cause of arts education and shares its benefits? 

Advocacy.

Calling school board members to ask them to support a specific, proposed measure in your district which would, for example, put an arts teacher in every elementary school? 

Lobbying.
People can lobby as an individual, as part of a group, or it can be part of their job. Organizations which do it (like certain nonprofits and issue-based organizations–like this one!) need to register with the State that these are part of their activities. But individuals speaking their own minds on a topic face no such criteria. 

People lobby their government as a way to engage in the democratic process, and it's a vital component. You elect your officials, but how do they know how people feel about a particular issue after that? How do they determine how to vote? Ideally, they look to their constituents and to knowledgeable parties with research about the issue and its possible impacts. It’s part of our first amendment right and an important part of the democratic process.

In that case, why does lobbying have such a bad reputation?

While individuals, groups, and organizations can lobby the government, so can corporations. And guess who has the most money and can hire professionals to lobby on their behalf, full time? Corporate lobbyists are typically working for what's best for big business, and this alone can be enough to give them a bad rap, because often those interests are drastically unaligned with what is better for families, workers, or the environment. There are also myriad ways that corporations can use this system to unduly influence legislators (if you want to find examples or know more, just query your favorite search engine).  

So why should YOU do it?

Because your interests are yours, not corporate. Tell your legislators and other elected officials what’s best for you, your family, and your community. Support your own values, by talking about what you want and need to the people who can help make it so. Align yourself with causes and organizations YOU care about, that don’t have big money backing them but which instead advocate, and yes, lobby, for change for good.

Lobbying is not a “one person, one vote” system, which can seem imbalanced. There's a growing amount of literature and introspection about how best to further represent the interests of people who don’t have the time or ability to show up for these and similar aspects of the democratic process. For example, if you’re a single parent who can’t afford childcare, what is the likelihood you’ll be able to go give feedback in person at a city council meeting? How can an elected official be reasonably sure that they are voting for their constituent’s best interest, when that person can’t tell them? That’s why grassroots organizations, nonprofits, and community organizations are a key part of building this bridge between citizen and representative. They have the ability to outreach directly to constituents to understand their needs and relay these to legislators. So use them, support them, and yes, if you are able, speak up on key issues!