This is a slightly deceptive screen as it not only deals with your player's overall training performance, but also gives crucial insight, from the point of view of your coaches, into the quality and future worth of your squad members.
You can use each of the four panels (info, schedule, training level and training progress) to not only see how you can improve a player, but also to analyse his worth to you and the wisdom of your selection and tactics.

Info
This is your absolutely key panel. It breaks down what schedule your player is on and tells you how fit your player is and how happy he feels about his training programme. These latter snippets should help you to judge whether you are pushing your player too hard, in which case you can modify the intensity of his training in the schedule panel. If a player is unhappy with his schedule it may pay to switch him to another programme or modify his regime on the training screen.
The New Position drop-down is a superb tool for developing your player within the framework of the squad, or for focusing a utility player on a specific position. It enables you to train a player to play in a new or specific position, but will inevitably damage his ability to play elsewhere.
Turning a left-back into a right-winger might take many years, and never fully succeed, so such dramatic changes should not be attempted. However, a DL/C could be trained into becoming an out-and-out centre-back with relatively little difficulty and could prove to be a far more useful member of your squad, uninhibited by his conflicting talents.
Without doubt the most useful aspect of this panel is the Coach report breakdown. Use the Coach drop-down button in the top right of the screen to alternate between your coaching staff and you'll get an overall picture of your player.
This report will be split into up to three main chunks: an analysis of the player's role in the team, an overview of the player's performance and a look ahead to how the player should fit into your future plans. Take heed of the advice here, which will often hint that the player is not being utilised correctly, and therefore may be playing out of position.
Crucially, the coach reports will give an indication of the player's attitude, suggesting that he may be giving less than 100% effort for the club or that he is not as committed to your cause as he should be. Such information should be taken into account alongside a player's character before the successful Football Manager tackles the issue head-on, rather than avoids it.

Schedule
This information corresponds with the data on the training screens and enables you to choose the intensity of your player's training. You'll not be able to modify the balance of his training here, however, only his workload and, in the info panel, his assigned schedule.
You can use this information in conjunction with the training level and progress boxes to analyse the effectiveness of the player's scheduled regime.

Training Level and Training Progress
These two graphs are closely connected. Training level shows the progress made this month in all of the training disciplines. Click on one of the bars and the training progress graph on the right will show a historical break down of the player's performance in the highlighted discipline over recent months.
This is a slightly time consuming, although extremely accurate, way of measuring the effects of a change in a training regime. If you set a player to train in a certain pattern one month and another the next, this direct comparison will enable you to see which is the most effective method for improving each individual.
For a more detailed breakdown of the statistical effects of your training you can examine the attributes chart.