They all relate to participation but they do imply different things. "Participate" is quite generic in that it just means someone is taking part or shares in an event - "to participate in a play." "Engage" tends to imply commitment and effort or a more personal level of participation, so that the one involved is interested - "to engage in conversation;" something "engaged her attention." It is also often used employment or use - "to engage a lawyer." "Involve" also goes deeper than "participate" in that what is involved is necessary and can refer to either persons or processes - "this involves commitment and effort." This is also the word often used for sensitive/difficult situations - "to involve in conflict;" "don't be involved in another's quarrel."