For opposite-sex couples, nothing has changed. The same-sex marriage decision does not affect your marriage in any way.
For same-sex couples, it will depend on if you are married and when you were married. If you were married before January 1, 2015, you will probably want to re-file your State taxes so that you can file as married. I would check with your accountant to see how far back you can re-file.
If you just got married this year, congratulations, and you will from this point on be filing joint Federal taxes and the corresponding State taxes.
For all same-gender couples, you will want to look at all of your financial and estate documents to see what you would like to redo. Things like the title on your house may need to be redone to reflect the marriage, depending on how the deed is currently worded. The same goes for things like living wills and wills. As it does make a difference legally to use the term spouse instead of partner or significant other, you may want to see about redoing those documents.
The big thing if for those couples with children. You are now able to see about second-parent adoptions or step-parent adoptions to give both parents the same legal status with the children.
This is just a very brief overview. Let us know if you have any questions.