A transitive verb is simply one that needs a direct object (a noun or a pronoun that the verb acts upon) to complete its thought. An intransitive one does not.
An example of a transitive verb is the English verb "to get" and one of its Spanish equivalents, obtener. If you were to use the verb by itself, such by saying "I get" in English or "obtengo" in Spanish, it is clear you aren't expressing a complete thought. There's a natural follow-up question here: What are you getting? ¿Qué obtengas? The verb simply isn't complete without an accompanying noun (or pronoun) to indicate what is being obtained: I am getting an error message. Obtengo un mensaje de error.
Another transitive verb is "to surpise" or its Spanish equivalent, sorprender. To express a complete thought, the verb must indicate who is surprised: It surprised me. Me sorprendió.
"To get," "to surprise," obtener and sorpender, then, are all transitive verbs. They must be used with an object.
Intransitive verbs are used without objects. They stand by themselves without acting on a noun or pronoun. Although they can be modified in meaning using adverbs or phrases, they cannot take a noun as an object. An example is the English verb "to flourish" and its Spanish equivalent, florecer. It doesn't make sense to flourish something, so the verb stands alone: The sciences flourished. Florecían las ciencias.
There are many verbs that can be used either transitively or intransitively. One example is "to study" or estudiar. You can use an object for a transitive usage (I am studying the book. Estudio el libro) or without an object for an intransitive usage (I am studying. Estudio.). "To write" and escribir can be used in exactly the same ways.
The distinctions between transitive and intransitive verbs usually don't give Spanish students a lot of trouble. Most of the time, when a transitive verb is used in English, you'll use a transitive one in Spanish. However, there are some verbs that can be used transitively in one language but not the other, or the opposite. That is one reason you may want to check the dictionary before you try using a verb in a way you haven't heard it before.
An example of a verb that can be used transitively in English but not Spanish is "to swim," as in "He swam the river." But the Spanish equivalent, nadar, can't be used in that way. While you can swim something in English, you can't nadar algo in Spanish. You'll need to recast the sentence: Nadó por el río.
The opposite can happen as well. In English, you can't sleep something, but in Spanish you can: La madre durmió al niño. The mother put the boy to sleep.