I was writing a lengthy post for this thread, clicked "New reply" aaand… ERROR. 8chan was down. Four months later, here we are again.
The oldest translations that are still common (though in actualized versions) are the Reina-Valera and the Torres Amat. The Reina-Valera is a protestant Bible and in a way the Spanish equivalent of the KJV: revered by some, considered outdated by the rest. It was first published by Reina in 1569 and revised by Valera in 1602 (hence the name Reina-Valera).
The Torres Amat Bible is a translation of the Vulgate, and it's usually full of notes and references to the Hebrew and Greek texts. The original one is from 1825 and while it's understandable (Spanish hasn't changed as much as English did in the last centuries) its language can be archaic, mainly cause it uses structures and vocabulary that was archaic in 1825 too. Modern versions, however, can be quite approachable.
Some common modern versions are:
-La Biblia de las Américas: Made in the US, it's my understanding that it's a quite popular version in the western side of the Atlantic. Apparently it's a quite literal translation of the Greek and Hebrew texts and can be quite dry.
-Biblia Latinoamericana: Catholic Bible post-Vatican II. Embraces the spirit of the council and was a quite polemic translation. Has a more south-american Spanish.
-Dios Habla Hoy: A Bible with simplified language. There are versions of several denominations, changing mostly the notes.
-Nueva Versión Internacional: Similar to the English NIV Bible, it's an evangelical Bible.
Other common versions, and the ones I'd recomend:
-Biblia de Navarra: Pretty new Catholic Bible, with Spanish text, translated from Greek and Hebrew, and Latin text next to it. It has notes and commentaries extracted from Church fathers and it's considered an excellent study Bible. However, can be pricier and hard to get.
-Biblia de Jerusalen: Considered by most the best Spanish translation, though the Biblia de Navarra has come to challenge that title. Sometimes accused of using a language that is only found in dictionaries, but that's an accusation I personally do not share. The notes and commentaries are magnificent. I'd say the Biblia de Navarra may be better if you're studying theology, but the Biblia de Jerusalen is the one for the average Catholic who's actually going to bother to read (and to try to understand) the Bible. For native speakers at least, it's the go-to version of the Bible.
-Sagrada Biblia (Versión de la Conferencia Episcopal Española): This is the official translation used in mass here in Spain. There's quite a broad consensus that it's not a particularly great translation. The language is clear, however, if a bit dry. The problem is that it's a compromise in trying to have modern, understandable language and be a faithful translation, and in all fairness, they achieved that, but at the cost of dry language and phrases sometimes a bit redundant or unnatural. For a native speaker, I'd recommend the Biblia de Jerusalen or the Biblia de Navarra, but maybe this one is better for a non-native speaker. You may not care that much about the literary quality of your Bible and just want a text you can understand.
This link is in Spanish, but has a list of all the Bible translations the Spanish Episcopal Conference has aproved:
https://es.wikipedia.org/wiki/Canon_b%C3%ADblico_de_la_Iglesia_cat%C3%B3lica#Versiones_en_espa%C3%B1ol_de_la_Biblia_cat%C3%B3lica
Check those Bibles out too. Many are South American translations, and if you're from the US, you may prefer a Bible that has a more Mexican or Argentinian Spanish.