Copy editing is usually provided by a publishing house, and is a thorough process whereby the content of the text is checked for consistency, accuracy, layout, and coherence. It may involve some rewriting. Many self-publishers do this themselves after writing a first draft, but can be ‘blind’ to errors as they are so familiar with the content.
For technical or specialised manuscripts it may be necessary to employ someone who is experienced in the relevant field.
Proofreading is the last step before publication, and involves close and careful scrutiny of the text in order to correct errors in spelling, punctuation, grammar and format. The content itself is assumed to be correct, and the proofreader works line by line to find and correct errors. For online manuscripts, this is done with Microsoft’s ‘Track Changes’, so that the author can clearly see the suggested corrections, and choose whether to accept or reject them. With the popularity of self-publishing growing day by day, this is a step which many new authors neglect, and which unfortunately can lead to poor reviews.
A beta reader is someone who reads your work (usually fiction), and gives feedback on it as a whole. They may look at issues such as characterisation, check for plot holes and so on. There are some offering this service for a low cost, or even free, and many are peer-to-peer arrangements. However, you should remember that they are not professional proofreaders or copy editors. Although an outside critique of your work may be enlightening and useful, it cannot take the place of having a professional check your work.