A long time ago this question was asked and /u/rusoved answered it with a quick snippet about the origins of the word.

As seen in that thread the word has a different meaning within Russian and within the Russian Empire. This actually ties back to the longer history of the Russian monarchy to before the Mongol Yoke and to the Kievan Rus era where the sovereign was known as the "Grand Prince." There would be lesser "Princes" who were rulers of separate regions, but under the Grand Prince. These separate rulers would be, in theory, be descendants of Rurik (the founder of Kievan Rus in the folklore, its all folklore, after all), but not direct descendants of the Grand Prince. The Kievan Rus system was a mess (the Rota system), and it requires graphs to better illustrate it, but it is important marker about how Prince was not just the same as the French or English royalty systems.

During the formation of the Russian Empire, this system changed as the Tsar and Emperor was the sole sovereign, and the Romanov Dynasty was the preeminent power, that would remain unchanged from the Time of Troubles to the Russian Revolution. This meant that the Tsar/Emperor(or Empress) would control the nobility completely, including their titles and land. This is important to note, because the nobility in Russia were in general, very weak in power, relying on the autocrat for their land which was granted by the autocrat, their titles which were also granted by the autocrat. The reasoning for this would deserve another question, as it relates to a greater history than would be necessary to cover for this one. Suffice to say, the Russian autocrat by the 18th and 19th centuries controlled the nobility and would "grant" titles to nobility for acts. Some may be heredity, but some were titular only.

This gets to the question at hand, who were the Princes? They were often the top of the nobility class in Russia, but with no relation to the autocrat, or even the Romanov family. One of the most famous Princes would be Prince Grigori Potemkin, the favorite of Empress Catherine II (the Great). He was born to a middle nobility, but through his work and recognition by Catherine he was granted the Title of Prince along with land and serfs in reward for his service to the crown. The other favorite of Catherine the Great was Orlov, who was also granted the title of Prince.

There was also the Princes as part of the "great houses" of Russia, and some of the other principalities of the Russian Empire. These were often families as descendants of the old system, as mentioned before, the Rurik family tree. If you were part of a family that could tie roots to the Rurik dynasty, then you were likely a Princely family. With the amount of the Princes that would have existed, it is feasible that there would be many destitute Prince families, as seen in War and Peace. The noble families would rely on the graciousness of the Romanov ruler, and if that relationship soured their ability to secure loans from the Crown could become troubled, and serfs could be levied away from their estates. Also, the Russian nobility in general, were poor in managing their estates and would frequently need funding from the Crown to maintain themselves. This further indebted the nobility to the autocrat, and kept the nobility weak and the autocrat in even higher absolute power.

The last bit, is what the titles of the actual Romanov are. The sovereign and autocrat, was the Tsar and/or Emperor(Empress), their direct, male heir, was the Tsesarevich. The other direct descendants of the Ruler, would be Grand Duke or Grand Duchess. We translate it this way, but the Russian is Великий князь, or Great Prince, as mentioned by /u/rusoved before. These would direct descendants of the Tsar, but could be the previous Tsar too. To better illustrate, the last Emperor and Tsar, Nicholas II had five children, Tsesarevich (and Grand Duke) Alexei, Grand Duchess Maria, Grand Duchess Olga, Grand Duchess Tatiana, and Grand Duchess Anastasia (yes, that one). Nicholas's brother Michael was also a Grand Duke, as well, and would remain so even after his brother's coronation.