Hispanic modernist literature, much like any modernist literature, can be seen as a rejection of realism. Realists try to write about things in the same way that a picture shows something: exactly as it is. Modernists weren't into this at all and instead looked inside themselves at their own emotions and thoughts. Realists wouldn't go for style points in their writing; modernists played with language and used weird words or styles to make their writing more lively and emotional. There is a strong focus on expression, beauty, and feeling. In general, modernists were rebellious and tried to create new things.
In terms of specifically Hispanic modernism, there are a few traits to look out for that are unique. The writing was often critical of society, governments, and the high class. There is often an element of spirituality to it, as well. Some of the writing is erotic, sometimes literally and other times figuratively. In general, Hispanic modernism was pretty open in terms of what was possible; Miguel de Unamuno, one of the largest figures of the movement, wrote in many different formats and styles. His poetry sometimes rhymed, and it sometimes didn't. Another common thread throughout the movement was politics that were often revolutionary-leaning. In general, though, the approach in Hispanic modernism was much like that of all other modernists.