Typically there are four career development paths.
First you want to become a more senior person within your own current specialty or a related specialty. So a junior dev might want to learn how to be senior dev or perhaps transition from being a C# developer to a data analyst or systems architect.
Next you might want to move into the management track. So you might currently be an accounting clerk but you really want to end up as the CFO (Chief Financial Officer) of a company.
Third you might want to change career tracks in your current industry or company. This would be something like a Meeting Planner wanting to change over to being a Human Resources specialist.
Fourth you might want to do something entirely different in a different industry such as training to be Nurse when you are currently an administrative assistant for a plumbing company.
Of course your boss is not likely to anything to help you in the last case. When you want to change both technical skill and industries, then you are simply announcing you are short-term and you will not get whatever help is available to others as it is a waste of company money and time.
Now the strategies for getting your boss to help you in the first three cases are:
First make sure your boss perceives you as doing your job well. If he thinks you have problems doing your current job, he is not going to help you get a better one. So don't talk to him first about your career aspirations but your current performance. Then fix anything he asks you to fix (whether you agree with it or not) before discussing future options.
Now you are ready to start preparing to discuss options. Before you do so, you need to develop a plan. You need to know the eventual goal and the intermediate positions you might need to get on the way there. The road from Junior dev to senior dev is fairly easy to figure out, but the road from accounting clerk to CFO is going to take multiple intermediate positions and probably some additional formal education.
Your immediate goal is to get an idea of the steps involved (the internet is a wonderful thing, use it to research the requirements at various levels including searching for the job you want on job sites to see what the qualifications are). Now focus on the next step you need to take. If possible, talk to people in that position and find out what qualifications they have and what their daily tasks are. Choose people who won't feel threatened by your questions. From your research, develop a list of things you need to get to be qualified for the next level and a plan for how you would like to get them. Don't ignore soft skills. These are often the difference between one level and the other.
The steps may now vary depending on whether you are following case 1 2 or 3. For cases 1 and 2 the initial steps are similar, you won't get to the management track until you reach the senior positions generally. So we start by discussing those first.
Now that you have a plan, you are ready to discuss steps with your current boss. If your boss is supportive of career development, you may be able to go to him and directly ask to be given the types of tasks you need to get to the next level. If he is not supportive of career development, simply volunteer for the tasks you would like to do to get qualified for the next level without discussing it as a career development move. Pay attention to upcoming projects and try to get assigned to the ones that further your plan.
Remember that a lot of the difference in the levels has to do with being able to work with less supervision, so concentrate on that as well.
One of the best ways to get the experience to move into management once you are a senior professional is to volunteer to start attending meetings for some of the projects you are working on instead of the manager to save him time. You can also ask for some of his responsibilities if he is going to be out of the office on vacation or long term leave of any kind in order to help out. Instead of telling him what you need him to do, save him time by creating the draft emails or giving him the information he needs to send something up the line rather than just saying you need him to do something. Now I will give you hint, it often easier to get what are really management tasks assigned to you when you have a poor manager. They don't want to do anything that takes extra effort, so you can often make them happy by doing things way beyond your level for them.
Now for case three, your immediate boss is not going to be much help even if he wants to be because he can't just assign you tasks outside his personal area of responsibility. What he can do is introduce you to the managers who do have those tasks, possibly get you assigned to some company-wide initiatives that are cross functional, or (and only if he is super supportive) lend you to the other department to do some tasks.
Some jobs lend themselves to this better than others. A lot of people like accounting clerks have gotten into programming jobs by doing special projects for their department that the programmers didn't have time to do. A developer who wants to be a business analysts can ask to help with the requirements for the next large project. A developer who want to get into Hr may ask to be assigned to the next project for the HR department both to start getting a functional understanding of what the department does and to make connections within the department.. These the types of things you can volunteer for and your boss can help you get assigned to.
Be aware that volunteering for tasks only works if you can keep up with the current workload in your actual job (which is possible if you are volunteering for tasks that will make something take less time in the long run like writing a system to automate an annoying manual process) or if it is possible for the boss to temporarily reassign you to that function. If he is going to have to give up someone to that project anyway (like a cross-functional project or a programming project for the department you would like to work in), your chances are better at getting assigned than if it is something totally unrelated to the workload.
If there is formal training you need, then research the possibilities and present a request for something specific. Your chances are better of getting it approved if it also meets the boss's needs. If there is an upcoming project that uses a new technology that you need to learn to get to where you want to be, then go to him with a training suggestion and volunteer for the project.
Look at your HR handbook to find out if there are things the company will reimburse you for such as getting a master's degree in your field or a related field. If this would further your own goals, find out how to apply and submit your application to your boss or whoever has to approve it rather than expecting someone to do it for you. Remember that sometimes these programs require a commitment to stay with the company so be aware of that before you take advantage of them. But by all means look at what the office will pay for and take advantage.
Another way to approach your boss to get help in personal development is to make a suggestion that might help everyone such as buying a technical library, getting an MSDN subscription or having series of lunch and learns or attending a training class and then coming back and presenting what you learned to the others.
Part of the plan to get to where you to where you want to go is to find the opportunities first and then ask your boss if you can get assigned to them rather than expecting him to come up with the plan or letting you know about the possibilities. Even good bosses will prefer this approach and bad bosses will almost certainly respond better to this approach.
So if you want to get those opportunities, you need to talk to people and pay attention to what is coming up and think about whether it will further your own plan to get assigned to it.
Make sure you look at what the boss will get from your suggestions and pitch them to him based on his needs to get the work done not on your desire to get promoted. These sales skills are ones you will need at anything past the most junior level anyway in most professions.
You will also want to discuss development during your performance appraisal and come prepared with suggestions for what you would like to get to move to the next level. Your boss will often have to put together a development plan for you and having you do the legwork and having concrete suggestions will make it easier for him and bosses like people who make things easier for them. Bosses tend to give more help to people who have made an effort to help themselves than people who sit there passively waiting for the boss to come up with a plan for how to get them promoted.