1080 = 1080 lines or pixels from the top to the bottom of the screen, 1920 lines or pixels from the left to the right
1080i = The image is interlaced which means the TV or monitor scans the odd 540 lines then comes back and does the even 540 lines (since a scan is from top to bottom). This means that as the picture is being reconstructed the picture elements which are moving are also changing location. This is not the best image quality if the source material was created, recorded, and played back as 1080p. No US broadcast TV channel can do better than 1080i. But high definition DVD formats can and computers and video games can.
1080p = Same number of lines as 1080i just progressive scan. Progressive scan means the TV monitor scans lines 1 to 1080 and then starts over with line 1. Best image quality. Better than US broadcast HDTV as noted above.
720p = Still progressive scan so goes from line 1 to line 720 and starts over with line 1. Less resolution since now it is 720 lines or pixels from top to bottom and 1280 lines or pixels from left to right. However this means that this signal can be broadcast as 720p on US HDTV channels.
What happens when 720p TV channel is displayed on 1080i or 1080p TV? It gets converted to display on the TV unless the TV has options to display in native 720p format. It is kind of complex but mathematical in nature (too hard to explain here).
Another kind of conversion process happens when 1080i or 1080p is displayed on 720p TV set.
Going from 1080i to 1080p or 1080p to 1080i is just a process of storing the image and playing back out in the opposite format.
All of this conversion process and the compression, encoding, decoding, and decompression of the bits in the digital signals is why there is a delay in digital TV (ATSC in the US, DVB-T in Europe) or digital satellite TV (small dishes) compared to analog TV (NTSC in the US, PAL in the UK, Secam in France, ...) or analog satellite TV (big dishes).
It is also a reason why surfing channels on a SDTV, HDTV, or digital cable box takes longer than analog channel surfing did. Too many formats for the computer chip inside the box to conform to and decode.