This is one of the most vexed issues facing the Latinx community, one that never seems to achieve resolution or consensus. Here is a quick primer on the issue.
The term "Hispanic" has long been used by government agencies to describe people who have a Spanish-speaking ancestor. As you might see, this already limits who can be counted among this group. Imagine, for instance, a person in the US whose grandparents emigrated from Spain in the early twentieth century. This person is Hispanic according to the definition, but they may have little else in common with other Hispanics who are not of European ancestry. In addition, during the civil rights movements in the US, Hispanic came to be seen as a government-aligned term, and so many communities rejected it, even to this day. We often see "Hispanic" because it is a part of the goverment-approved language.
Generally speaking, "Latino" was much more preferred, even though many groups thought (and still think) that it is too generalizing a term. It's a "pan-" term that seeks to smooth over the differences and distinctions among Latino groups. Many see this as an erasure of culture and identity. The support for this overarching term comes from the idea that "Latino" not only creates solidarity for the group, but it enhances its visibility and political power. It is best to think of "Latino" and "Hispanic" as umbrella terms under which there are more nuanced identity-categories such as Mexican American (sometimes seen in its more political activist version as "Chicano," Cuban-, Honduran-, Equadorian-, Peruvian-American, and more).
You may wonder why "Latino" isn't sufficient or good enough. As Spanish is a language that has gender designations for things that may not have gender in actuality, when brought into English usage, words carry greater gender implications. "Latino" is masculine, and in Spanish we see the term used even in groups that are comprised of men, women, and non-gender peoples. For a time, academics used various methods to get around this issue of gender. They used "Latin@" or even "Latina/o" to denote a more inclusive term. With the acknowledgment of non-binary genders, even "Latina/o" is not as inclusive as it first seems. Enter the "x."
The "x" in "Latinx" is a kind of place holder to be filled in by the person who reads or hears it. It is an open-ended term that signals inclusivity and invitation. Though it is not universally agreed upon, more and more people are using the term.
So how is it pronounced? The majority of people (at the moment) pronounce it "La-TEEN-ex," while almost the same amount of people pronounce it "Latin-ex." See which one works for you.
The Latinx Cultural Center uses the "x" to signify its openness to diverse identities within the community.