10 characteristics, eh? Modern Philosophy is a huge blanket term referring to roughly everything after Descartes. But it's good for a student to review his subject field. Here's a smattering of trends, subject and and features of modern philosophy I can think of. I'll try not to be too 20th century biased, and probably fail.
1. Faction in the 20th Century - Analytic Philosophy vs. Continental Philosophy
It should be well known to a modern student of philosophy that house currently stands somewhat divided on itself. In England and America, analytic philosophy in the tradition of Russell and others dominates. It has essentially taken over. But in the 20th century we saw the rise of philosophers rejecting "scientism", particularly in continental Europe. Names like Husserl, Kierkegaard and Sartre stand out prominently as examples of philosophers not plugged into the American/English tradition.
2. Modern Logic
Between great thinkers like Frege, Gödel, Tarski, Kripke and Quine, in the 20th century we developed an unparalleled understanding of philosophical and mathematical logic. Model theory remains a hot topic in mathematical and philosophical circles. A basic understanding of the nature of incompleteness and completeness is becoming standard for student of philosophy and computer science. With the advance of logic philosophy over all has become a more formal discipline.
3. Thinking in Terms of Language
Much of modern linguistics has philosophical fathers. Frege and Grice spring to mind, a strong case could also be made for Tarski. But the architect of a takeover of language analysis in the 20th century was Wittgenstein. His Philosophical Investigations lead philosophers to the idea that by analyzing so called language games, we could solve paradoxes and understand the world. This was something of a pre-occupation of philosophers in the 20th century.
4. Epistemology - Empiricism vs. Rationalism before the 20th century
The early modern philosophers were deeply concerned with questions of epistemology. History class usually breaks them up into two approaches. One approach put "self-evident" truths into the center stage. This approach is called rationalism. The other approach put sense data as the primary source of knowledge. This approach is known as empiricism. The subject of many treatises has been based on argument for one side against another. Some of the great rationalist of history are Descartes and Leibniz. Some of the great empiricists are Hume and Locke. It is worth mentioning that Kant argued against rationalism on the basis of arguments with equal a priori support, which he dubbed "antinomies".
5. Philosophy of Government
The idea of what makes a good government takes center stage from the enlightenment all the way to modern day. Hobbes, Rousseau, Locke, John Stuart Mill, Karl Marx, Robert Nozick and John Rawls are just a few names in a long, rich history of political philosophers. It might be worth mentioning that enthusiastic readers of author Ayn Rand, who go under the self-proclaimed name of Objectivists count themselves as political philosophers (as well as philosophers of epistemology and ethics). As of right now, most philosophy departments ignore Objectivists.
6. Science - What is it? What is Causality? What is a Scientific Explanation?
In the 20th century, the philosopher Karl Popper raised the question "What makes something scientific?" and argued that Freudian psychology and Marxism are not scientific modes of thought, but that relativity, which were cutting edge at the time, is. His arguments were based on a principle of "falsifiability", namely that in order for some hypothesis to be scientific, it must be falsifiable. Other philosophers have grappled with the subject of causality, the earliest account I can think of in the 20th century belonging to Hume. Still others such as Hempel grappled with the notion of what is an explanation, giving rise to the idea of a Deductive-Nomological account of scientific explanation. Modern philosophers of science include Nancy Cartwright and Bas van Fraassen.
7. Modern Epistemology - Justified True Belief
The subject of epistemology has been hot and remains hot throughout all of Modern philosophy. I have already mentioned some of the classical approaches to epistemology, namely rationalism and empiricism. The modern approach barrows from the seminal critique of modern epistemology by Edmund Gettier. Read up more on "Gettier cases" if you are curious. Much of epistemology is devoted to the question of "What makes a belief justified?"; but also shares topics in ontology (namely the question, "What is true?") and philosophy of mind (namely the subject, "What is a belief?"). A pioneer in our modern theory of knowledge would be W.V. Quine, who suggests that beliefs form webs with peripheries we are more likely to abandon in the face of conflicting evidence and cores we would essentially never abandon.
8. Ethics - Cognitive ( Utility vs. Obligation) vs. Non-Cognitive
In the 20th century the field of Meta-Ethics was formed, to answer the question "What is ethics anyway?" An early name in this subject was G.E. Moore. Essentially the question is split between ideas that ethical truths can be discerned objectively somehow, versus the idea that ethical statements mean "something else". An early Cognitivist contrast to Moore was the philosopher W.D. Ross, who gives the idea of candidate duties one must decide between in situations of seeming conflict of duty, or as he calls them prima facia duties. The Non-Cognitivist philosopher Ayer did not believe that there was any way of resolving ethical disputes. The Non-Cognitivist philosopher Stevenson thought that ethical statements were supposed to evince another person of some position.
9. Philosophy of Mind
Another subject and perennial question of modern philosophy is "What is the thought?" The discussion almost always starts with Descartes Meditations. The view is typically split between Materialism, which holds that thoughts can be explained in terms of matter, and Dualism, which holds that thoughts can be explained in terms other worldly matter. Almost all 20th century analytic philosophers reject this view.
10. Rationality and Economy
Ever since Adam Smith, we have been interested in asking what motivates people in economies, and what is best. A closely related question is "What is rationality". I will be honest, this subject matter is the focus of my study. Great reading can be found classically in Malthus and Hobbes and Smith, great modern reading would be in Von Neumman & Morgenstern, Luce & Raiffa, Kripke, David Lewis, Auman, Kreps, Rubinstein, Amartia Sen, Schelling and Kahneman to mispell a few names off the top of my head.
Not an exhaustive list by any measure. But I did take some time to compile it...