The word queer, when used to describe people, was originally used only as a derogatory term for gay men or anyone who read as homosexual (that is effeminate or otherwise different or odd). Calling someone queer was very much a negative thing, and the term is still used today by some heterosexuals as an insult - and it's still considered by many (gay and not) as offensive.
But in the late 1980s, the word queer began to be used by people to describe themselves, as a way of "taking back" the word and transforming its meaning. Here, queer was used proudly to distinguish oneself from heteronormative and gender normative conventional ways of thinking and behaving. Queer was both a political stance as well as a description of who one might choose to have sex with, to love, to be in a relationship with, and more.
Today, people use queer in a variety of different ways. Usually, it's used by people who identify as something other than heterosexual, although it can also be used by people who identify as trans and genderqueer.
Some use it to describe their sexual identity, one that rejects conventional ideas of gender and sexual orientation binaries that are fixed and always either/or. While someone might be gay, lesbian, or bisexual, and still feel very much in tune with conventional ideas about sexuality and gender, people who are queer are making a point of saying that mainstream and conventional modes of sexuality and gender are too rigid, and continue to center the experience of heterosexuality. Queer people, for example, would usually challenge the idea of compulsory monogamy, regardless of the genders or orientations of people in the relationship.
For many, the identity of queer also has a political element. Being queer for these people means being left of center, and often includes being concerned with workers rights and with the impact of systemic oppression on marginalized groups - whether it's based on gender, orientation, class, race, or other identities.
Some people use queer to describe a broader range of sexual interests, or to indicate that they are open to sexual alternatives (like BDSM, fetishes) and relationship alternatives (for example non-monogamy and polyamory). People sometimes identify as queer as a way of including in their world people who don't feel like they fit any of the binaries - for example, some trans people, or people who identify as two-spirit, who don't identify as either just man or woman, or people who identify as pansexual or asexual. These are all very different kinds of identities, but they all are marginalized, and for some people, calling themselves queer is a way of including more people in their community.
Language is always evolving, and queer is one of those words whose meaning seems to change a lot. It's still a word that can hurt people, and it isn't recommended to use for describing someone else, unless they themselves have used it. And even then, it's important to remember that others may find it offensive; using it is a choice one makes.