From Wikipedia:

720p versus 1080i
Some United States broadcasters use 720p60 as their primary high-definition format; others use the 1080i standard. While 720p presents a complete 720 line frame to the viewer between 24 to 60 times each second (depending on the format), 1080i presents the picture as 50 or 60 partial 540 line "fields" (24 complete 1080-line fields, or "24p" is included in the ATSC standard though) which the human eye or a deinterlacer built into the display device must visually and temporally combine to build a 1080 line picture - in CRT type display. To get all 1080 interlaced lines to appear on the screen at the same time on a progressive high-definition display, the processor within the HD set has to weave together both 540-line segments to form the full-resolution frame. It does so by holding the first field in its memory, receiving the next field, then electronically knitting the two fields together. The combined fields are displayed at once as a complete 1080p frame. The main tradeoff between the two is that 1080i may show more detail than 720p for a stationary shot of a subject at the expense of a lower effective refresh rate and the introduction of interlace artifacts during motion. 720p is used by ABC and ESPN because the smoother image is desirable for fast-action sports telecasts. Fox Broadcasting Company uses the tagline "the nation's finest high-definition standard" in advertising its 720p programming.

1080i vs. 1080p
The progressive-scan versions of the 1080-line resolution is known as 1080p. Current digital television broadcast systems and standards are not equipped for 1080p50/60 transmission. Also, the majority of consumer televisions offered for sale are currently not equipped to receive or decode a 1080p signal at any frequency. It is less bandwidth-intensive to broadcast a film at 1080p24 than 1080i60, since 20% less data would be transferred. In addition, when the source material is 24 frames per second, as are most films, it would be easy to convert a 1080p24 broadcast to an NTSC 1080i60 format using a 3:2 pulldown process (see telecine). Moreover, displaying a p24 broadcast on an i50 system (such as PAL) requires the speed of video and audio be increased by over 4% (to 25 frames per second).
Therefore a hybrid is often used for films and the like: the frames (25 or 30 per second) are segmented into two interlaced fields with equal time index (psf, progressive in or with segmented frames). The deinterlacer has to perform a simple weave only. This ensures compatibility with 1080i50/60 with only little less coding efficiency than 1080p25/30 and half the bandwidth requirement of 1080p50/60, but the SDTV problems of PAL speed-up and Telecine judder remain.
It is less efficient to transmit any signal with a frame rate significantly higher than that of its source, which is 24fps for films; however, frame differencing in the compression standards used for HDTV transmission greatly reduce the wasted bandwidth in these cases.

1080p = 1920 × 1080
1080p is the shorthand name for a category of video modes. The number 1080 represents 1,080 lines of vertical resolution,[1] while the letter p stands for progressive scan or non-interlaced. 1080p is considered a HDTV video mode. The term usually assumes a widescreen aspect ratio of 16:9, implying a horizontal (display) resolution of 1920 dots across and a frame resolution of 1920 × 1080 or about 2.07 million pixels. The frame rate in hertz can be either implied by the context or specified after the letter p (such as 1080p30, meaning 30 frames per second).
1080p is sometimes referred to in marketing materials as "True High-Definition" or "Full High-Definition". 1080p is currently the digital standard for filming digital motion pictures. The best technology commercially available is 1080p, soon to be superseded by 2160p.

720p
720p is the shorthand name for a category of HDTV video modes. The number 720 stands for 720 lines of vertical display resolution, while the letter p stands for progressive scan or non-interlaced. When broadcast at 60 frames per second, 720p features the highest temporal (motion) resolution possible under the ATSC standard. Progressive scanning reduces the need to prevent flicker by filtering out fine details, so spatial (sharpness) resolution is much closer to 1080i than the number of scan lines would suggest.

1080i = field resolution of 1920 × 1080 /2
1080i is shorthand name for a category of video modes. The number 1080 stands for 1080 lines of vertical resolution, while the letter i stands for interlaced or non-progressive scan. 1080i is considered to be an HDTV video mode. The term usually assumes a widescreen aspect ratio of 16:9, implying a horizontal resolution of 1920 pixels and a frame resolution of 1920 × 1080 or about 2.07 million pixels, and a field resolution of 1920 × 1080 / 2 (because it's interlaced) or about 1.04 million pixels.