简单说Checked就是在compile阶段会检查，必须用try catch或者throws处理，比如FileNotFoundException

Unchecked就是compile阶段不会检查，比如ArithmeticException 除法（除以0）

In Java, there two types of exceptions:

1) Checked: are the exceptions that are checked at compile time. If some code within a method throws a checked exception, then the method must either handle the exception or it must specify the exception using throws keyword.

For example, consider the following Java program that opens file at locatiobn “C:\test\a.txt” and prints first three lines of it. The program doesn’t compile, because the function main() uses FileReader() and FileReader() throws a checked exception FileNotFoundException. It also uses readLine() and close() methods, and these methods also throw checked exception IOException

import java.io.*;
 
class Main {
    public static void main(String[] args) {
        FileReader file = new FileReader("C:\\test\\a.txt");
        BufferedReader fileInput = new BufferedReader(file);
         
        // Print first 3 lines of file "C:\test\a.txt"
        for (int counter = 0; counter < 3; counter++) 
            System.out.println(fileInput.readLine());
         
        fileInput.close();
    }
}

Output:

Exception in thread "main" java.lang.RuntimeException: Uncompilable source code - 
unreported exception java.io.FileNotFoundException; must be caught or declared to be 
thrown
	at Main.main(Main.java:5)

To fix the above program, we either need to specify list of exceptions using throws, or we need to use try-catch block. We have used throws in the below program. Since FileNotFoundException is a subclass of IOException, we can just specify IOException in the throws list and make the above program compiler-error-free.

import java.io.*;
 
class Main {
    public static void main(String[] args) throws IOException {
        FileReader file = new FileReader("C:\\test\\a.txt");
        BufferedReader fileInput = new BufferedReader(file);
         
        // Print first 3 lines of file "C:\test\a.txt"
        for (int counter = 0; counter < 3; counter++) 
            System.out.println(fileInput.readLine());
         
        fileInput.close();
    }
}

Output: First three lines of file “C:\test\a.txt”

2) Unchecked are the exceptions that are not checked at compiled time. In C++, all exceptions are unchecked, so it is not forced by the compiler to either handle or specify the exception. It is up to the programmers to be civilized, and specify or catch the exceptions.
In Java exceptions under Error and RuntimeException classes are unchecked exceptions, everything else under throwable is checked.

                   +-----------+
		   | Throwable |
                   +-----------+
                    /         		   /                     +-------+          +-----------+
          | Error |          | Exception |
          +-------+          +-----------+
	   /  |  \           / |                 \________/	  \______/    	 			                +------------------+
	unchecked	 checked	| RuntimeException |
					+------------------+
					  /   |    |      					 \_________________/
					   
					   unchecked

Consider the following Java program. It compiles fine, but it throws ArithmeticException when run. The compiler allows it to compile, because ArithmeticException is an unchecked exception.

class Main {
   public static void main(String args[]) {
      int x = 0;
      int y = 10;
      int z = y/x;
  }
}

Output:

Exception in thread "main" java.lang.ArithmeticException: / by zero
	at Main.main(Main.java:5)
Java Result: 1

Why two types?
See Unchecked Exceptions — The Controversy for details.

Should we make our exceptions checked or unchecked?
Following is the bottom line from Java documents
If a client can reasonably be expected to recover from an exception, make it a checked exception. If a client cannot do anything to recover from the exception, make it an unchecked exception

http://www.geeksforgeeks.org/checked-vs-unchecked-exceptions-in-java/

Checked vs Unchecked Exceptions in Java