UPDATE

This answer is rather old, and so describes what was 'good' at the time, which was smart pointers provided by the Boost library. Since C++11, the standard library has provided sufficient smart pointers types, and so you should favour the use of std::unique_ptr, std::shared_ptr and std::weak_ptr.

There was also std::auto_ptr. It was very much like a scoped pointer, except that it also had the "special" dangerous ability to be copied — which also unexpectedly transfers ownership.
It was deprecated in C++11 and removed in C++17, so you shouldn't use it.

std::auto_ptr<MyObject> p1 (new MyObject());
std::auto_ptr<MyObject> p2 = p1; // Copy and transfer ownership. 
                                 // p1 gets set to empty!
p2->DoSomething(); // Works.
p1->DoSomething(); // Oh oh. Hopefully raises some NULL pointer exception.

OLD ANSWER

A smart pointer is a class that wraps a 'raw' (or 'bare') C++ pointer, to manage the lifetime of the object being pointed to. There is no single smart pointer type, but all of them try to abstract a raw pointer in a practical way.

Smart pointers should be preferred over raw pointers. If you feel you need to use pointers (first consider if you really do), you would normally want to use a smart pointer as this can alleviate many of the problems with raw pointers, mainly forgetting to delete the object and leaking memory.

With raw pointers, the programmer has to explicitly destroy the object when it is no longer useful.

// Need to create the object to achieve some goal
MyObject* ptr = new MyObject(); 
ptr->DoSomething(); // Use the object in some way
delete ptr; // Destroy the object. Done with it.
// Wait, what if DoSomething() raises an exception...?

A smart pointer by comparison defines a policy as to when the object is destroyed. You still have to create the object, but you no longer have to worry about destroying it.

SomeSmartPtr<MyObject> ptr(new MyObject());
ptr->DoSomething(); // Use the object in some way.

// Destruction of the object happens, depending 
// on the policy the smart pointer class uses.

// Destruction would happen even if DoSomething() 
// raises an exception

The simplest policy in use involves the scope of the smart pointer wrapper object, such as implemented by boost::scoped_ptr or std::unique_ptr.

void f()
{
    {
       std::unique_ptr<MyObject> ptr(new MyObject());
       ptr->DoSomethingUseful();
    } // ptr goes out of scope -- 
      // the MyObject is automatically destroyed.

    // ptr->Oops(); // Compile error: "ptr" not defined
                    // since it is no longer in scope.
}

Note that std::unique_ptr instances cannot be copied. This prevents the pointer from being deleted multiple times (incorrectly). You can, however, pass references to it around to other functions you call.

std::unique_ptrs are useful when you want to tie the lifetime of the object to a particular block of code, or if you embedded it as member data inside another object, the lifetime of that other object. The object exists until the containing block of code is exited, or until the containing object is itself destroyed.

A more complex smart pointer policy involves reference counting the pointer. This does allow the pointer to be copied. When the last "reference" to the object is destroyed, the object is deleted. This policy is implemented by boost::shared_ptr and std::shared_ptr.

void f()
{
    typedef std::shared_ptr<MyObject> MyObjectPtr; // nice short alias
    MyObjectPtr p1; // Empty

    {
        MyObjectPtr p2(new MyObject());
        // There is now one "reference" to the created object
        p1 = p2; // Copy the pointer.
        // There are now two references to the object.
    } // p2 is destroyed, leaving one reference to the object.
} // p1 is destroyed, leaving a reference count of zero. 
  // The object is deleted.

Reference counted pointers are very useful when the lifetime of your object is much more complicated, and is not tied directly to a particular section of code or to another object.

There is one drawback to reference counted pointers — the possibility of creating a dangling reference:

// Create the smart pointer on the heap
MyObjectPtr* pp = new MyObjectPtr(new MyObject())
// Hmm, we forgot to destroy the smart pointer,
// because of that, the object is never destroyed!

Another possibility is creating circular references:

struct Owner {
   std::shared_ptr<Owner> other;
};

std::shared_ptr<Owner> p1 (new Owner());
std::shared_ptr<Owner> p2 (new Owner());
p1->other = p2; // p1 references p2
p2->other = p1; // p2 references p1

// Oops, the reference count of of p1 and p2 never goes to zero!
// The objects are never destroyed!

To work around this problem, both Boost and C++11 have defined a weak_ptr to define a weak (uncounted) reference to a shared_ptr.

Throwing an exception out of a destructor is dangerous.
If another exception is already propagating the application will terminate.

#include <iostream>

class Bad
{
    public:
        // Added the noexcept(false) so the code keeps its original meaning.
        // Post C++11 destructors are by default `noexcept(true)` and
        // this will (by default) call terminate if an exception is
        // escapes the destructor.
        //
        // But this example is designed to show that terminate is called
        // if two exceptions are propagating at the same time.
        ~Bad() noexcept(false)
        {
            throw 1;
        }
};
class Bad2
{
    public:
        ~Bad2()
        {
            throw 1;
        }
};


int main(int argc, char* argv[])
{
    try
    {
        Bad   bad;
    }
    catch(...)
    {
        std::cout << "Print This\n";
    }

    try
    {
        if (argc > 3)
        {
            Bad   bad; // This destructor will throw an exception that escapes (see above)
            throw 2;   // But having two exceptions propagating at the
                       // same time causes terminate to be called.
        }
        else
        {
            Bad2  bad; // The exception in this destructor will
                       // cause terminate to be called.
        }
    }
    catch(...)
    {
        std::cout << "Never print this\n";
    }

}

This basically boils down to:

Anything dangerous (i.e. that could throw an exception) should be done via public methods (not necessarily directly). The user of your class can then potentially handle these situations by using the public methods and catching any potential exceptions.

The destructor will then finish off the object by calling these methods (if the user did not do so explicitly), but any exceptions throw are caught and dropped (after attempting to fix the problem).

So in effect you pass the responsibility onto the user. If the user is in a position to correct exceptions they will manually call the appropriate functions and processes any errors. If the user of the object is not worried (as the object will be destroyed) then the destructor is left to take care of business.

An example:

std::fstream

The close() method can potentially throw an exception. The destructor calls close() if the file has been opened but makes sure that any exceptions do not propagate out of the destructor.

So if the user of a file object wants to do special handling for problems associated to closing the file they will manually call close() and handle any exceptions. If on the other hand they do not care then the destructor will be left to handle the situation.

Scott Myers has an excellent article about the subject in his book "Effective C++"

Edit:

Apparently also in "More Effective C++"
Item 11: Prevent exceptions from leaving destructors