One way of providing Open Access is to publish in an Open Access Journal. These journals make their articles available for free by funding their publication services in other ways than through end-user subscriptions. A lot of journals fund their workings by charging Article Processing Charges (APCs), although not all publishers do and there exist huge variations in the actual amount of the APC.
A lot of traditional journals also offer the possibility of making individual articles Open Access upon payment of an APC (in this case they are called hybrid journals). Unfortunately, while the individual article thus becomes freely available, the journal as a whole remains closed access.
APCs can be included in the costs of research funding, so the money for access comes through the research funder, rather than through the library budget. Of course, the initial source of the money is often the same (from government funding).
There is a growing number of Open Access Journals; most disciplines are now represented. A comprehensive overview is provided by the Directory of Open Access Journals, DOAJ.
Some Open Access Journals offer you the option of archiving the underlying data of an article you submit as well, and there are even journals who only publish datasets and their metadata.