Spanish speakers from Madrid, Mexico City and Buenos Aires understand each other without trouble. The differences come down to a handful of sounds.
1. C and Z: "th" or "s"
This is the big one. In most of Spain, z and c before e or i sound like the "th" in "think". In Latin America (and the Canary Islands) they sound like s.
Switch the accent above between Spain and a Latin American option, then play: gracias, cerveza, zapato, Barcelona, Ibiza.
It isn't a lisp. Spain also has a normal s sound in words like casa, and the two are kept apart: in Spain casa (house) and caza (hunt) sound different, while in Latin America they sound the same. The popular story about a king with a lisp is a myth.
2. The j
The j and soft g are raspy in Spain, made far back in the throat. In Mexico, Central America and the Caribbean they're softer, closer to an English h. Compare jamón ibérico and ojo in both accents. More in the J and G guide.
3. The s at the end of a syllable
In the Caribbean, much of coastal Latin America, Argentina and southern Spain, an s before a consonant or at the end of a word often softens into an h or disappears: más o menos can sound like "mah o meno". Mexico City, Bogotá and central Spain keep a clear s. If you're learning, pronounce every s.
4. LL and Y in Argentina
In Buenos Aires and Montevideo, ll and y sound like "sh": "calle" sounds like KAH-sheh. Elsewhere they sound like the y in "yes". See the LL and Y guide.
Which one should you learn?
Either is correct, so pick the one you'll actually hear. In the US, Latin American (especially Mexican) pronunciation is the most useful. For travel or study in Spain, go with the Spain accent. Mixing a little is normal; what matters is being consistent with the th/s choice.