In Apex Exception are handles using try, catch and finally.

try – block of code where an error can occur.

catch – block handles particular type of exception. A single try block can have zero or more exception block.  Each catch block must have unique exception type. Once a particular exception type is caught is one catch block, the remaining catch block will not execute.

finally – this block of code execute always and used for clean up code or sending email. A try block can have zero or one finally block.

try {
// Try block
***Code Block***
} catch (exceptionType variableName) {
// Initial catch block.
// At least the catch block or the finally block must be present.
code_block
} catch (Exception e) {
// Optional additional catch statement for other exception types.
// Note that the general exception type, 'Exception',
// must be the last catch block when it is used.
code_block
} finally {
// Finally block.
// At least the catch block or the finally block must be present.
code_block
}

At least a catch block or a finally block must be present for a try block.   catch  block is not mandatory for a try  block.

Below syntax both are valid and will work fine –

Try-Catch Block

try {
***code_block***
} catch (exceptionType variableName) {
code_block
}
// Optional additional catch blocks

Try-Finally Block

try {
***code_block***
} finally {
code_block
}

The finally block always executes regardless of exception is thrown, and even if no exception is thrown.  Finally block executed after the exception caught.

Built-In Exceptions

Apex provides a different type of built-in exception types that the runtime engine throws if errors are encountered during execution. All exceptions support built-in methods for returning the error message and exception type.

Exception	Description
AsyncException	Any problem with an asynchronous operation, such as failing to enqueue an asynchronous call.
CalloutException	Any problem with a Web service operation, such as failing to make a callout to an external system.
DmlException	Any problem with a DML statement, such as an insert statement missing a required field on a record.
EmailException	Any problem with email, such as failure to deliver. For more information, see Outbound Email.
ExternalObjectException	Any problem with external object records, such as connection timeouts during attempts to access the data that’s stored on external systems.
InvalidParameterValueException	An invalid parameter was supplied for a method or any problem with a URL used with Visualforce pages. For more information on Visualforce, see the Visualforce Developer’s Guide.
LimitException	A governor limit has been exceeded. This exception can’t be caught.
JSONException	Any problem with JSON serialization and deserialization operations. For more information, see the methods of System.JSON, System.JSONParser, andSystem.JSONGenerator.
ListException	Any problem with a list, such as attempting to access an index that is out of bounds.
MathException	Any problem with a mathematical operation, such as dividing by zero.
NoAccessException	Any problem with unauthorized access, such as trying to access an sObject that the current user does not have access to. This is generally used with Visualforce pages. For more information on Visualforce, see the Visualforce Developer’s Guide.
NoDataFoundException	Any problem with data that does not exist, such as trying to access an sObject that has been deleted. This is generally used with Visualforce pages. For more information on Visualforce, see the Visualforce Developer’s Guide.
NoSuchElementException	This exception is thrown if you try to access items that are outside the bounds of a list. This exception is used by the Iterator next method. For example, ifiterator.hasNext() == false and you call iterator.next(), this exception is thrown. This exception is also used by the Apex Flex Queue methods and is thrown if you attempt to access a job at an invalid position in the flex queue.
NullPointerException	Any problem with dereferencing null, such as in the following code: String s; s.toLowerCase(); // Since s is null, this call causes // a NullPointerException
	Any problem with SOQL queries, such as assigning a query that returns no records or more than one record to a singleton sObject variable.
RequiredFeatureMissing	A Chatter feature is required for code that has been deployed to an organization that does not have Chatter enabled.
SearchException	Any problem with SOSL queries executed with SOAP API search() call, for example, when the searchString parameter contains less than two characters. For more information, see the SOAP API Developer Guide.
SecurityException	Any problem with static methods in the Crypto utility class. For more information, see Crypto Class.
SerializationException	Any problem with the serialization of data. This is generally used with Visualforce pages. For more information on Visualforce, see the Visualforce Developer’s Guide.
SObjectException	Any problem with sObject records, such as attempting to change a field in an updatestatement that can only be changed during insert.
StringException	Any problem with Strings, such as a String that is exceeding your heap size.
TypeException	Any problem with type conversions, such as attempting to convert the String ‘a’ to an Integer using the valueOf method.
VisualforceException	Any problem with a Visualforce page. For more information on Visualforce, see the Visualforce Developer’s Guide.
XmlException	Any problem with the XmlStream classes, such as failing to read or write XML.

Exception Methods

Name	Return Type	Description
getCause	Exception	Returns the cause of the exception as an exception object.
getLineNumber	Integer	Returns the line number from where the exception was thrown.
getMessage	String	Returns the error message that displays for the user.
getStackTraceString	String	Returns the stack trace as a string.
getTypeName	String	Returns the type of exception, such as DmlException, ListException, MathException, and so on.