The differences between the Spanish of Spain and the Spanish of Latin America can be compared to the differences between British English and American English. People from throughout the Spanish-speaking world can communicate with other as easily as people throughout the English-speaking world can.
There are differences, more visible in the spoken language than in writing, but they aren't so extreme. In Latin American Spanish the direct loanwords from English are more frequent, without translating or adapting the spelling to the traditional norms. The most notorious example is the use of the word email or e-mail in Latin American instead of the more literal translation, correo electrónico, that is used in Spain . These differences are evident especially in recently adopted technical terms. In Latin America they say la computadora while in Spain it's el ordenador, and each of the two words sounds foreign in the region where it is not used.
Also, while it's easy to think of Latin American Spanish as one entity, you should keep in mind that the language has variants in the diverse zones where it is spoken. These differences are called regional variants or dialects. In Latin American this tendency toward differentiation is almost imposed by the very magnitude of the territory.
But again, the differences aren't so extreme that they prevent communication. Latin Americans watch movies from Spain, and Spaniards watch Latin American telenovelas (soap operas) You might want to avoid slang or extreme colloquialisms, but standard Spanish is understood anywhere in the Spanish-speaking world.
Here, however, we describe some of the differences you may find:

Pronunciation: One of the main differences is that many Spaniards often pronounce the z and the c before i or e like the "th" in "thin," while many Latin Americans pronounce it the same as the s. Also, speakers in some areas ( Argentina in particular) often pronounce the ll and y like the "s" in "measure." In some areas, you will hear speakers drop s sounds, so está sounds like etá. In some areas, the j sounds like the "ch" in "loch" (difficult for many native English speakers to master), while in others it sounds like the English "h." In some areas, the l and the r at the end of a word sound alike.

Grammar: Two of the biggest differences, each worth a lesson in itself, are the leísmo of Spain and the use of the pronoun vos in some areas instead of tú.

Vocabulary: Other than slang, probably the biggest class of vocabulary differences you'll come across is in the use of suffixes. A lápiz is a pencil or crayon everywhere, but a lapicero is a pencil holder in some areas, a mechanical pencil in others. Of course, every area also has its own words. For example, a Chinese restaurant in Chile or Peru is called a chifa, but you won't run across that word in many other places.