I live in Seattle. Do I have a better chance of getting into Seattle University compared to someone from, say New York or someplace out of state? How about other state schools like the University of Washington or Washington State?
In general, in-state students are admitted at higher rates than out-of-state students at public colleges. This is often a result of state laws that dictate quotas for the percentage of in-state students in each class. Public universities are intended primarily to serve the residents of the states where they are located. This trend has changed at a few schools (particularly state schools in California like UCLA) because out-of-state students pay higher tuition and bring in more revenue, but it appears that Washington's public universities still offer admissions advantages to in-state applicants.
Since Seattle University is a private school, you won't have an advantage there. However, for University of Washington and Washington State, your in-state residency will give you a bit of an edge. Here's a quote from an article about admissions at University of Washington:
"'The good news for Washington residents is that they do not compete with nonresidents in the admissions process,' said UW admissions director Paul Seegert. 'In fact, although residents comprise only one-third of the applications, about two-thirds of the enrollment space is reserved for them.'”