1080i and 1080p are both High Definition display formats for HDTVs. 1080i and 1080p signals actually contain the same information. Both 1080i and 1080p represent a 1920x1080 pixel resolution (1,920 pixels across the screen by 1,080 pixels down the screen). The difference between 1080i and 1080p is in the way the signal is sent from a source component or displayed on an HDTV screen.
In 1080i each frame of video is sent or displayed in alternative fields. The fields in 1080i are composed of 540 rows of pixels or lines of pixels running from the top to the bottom of the screen, with the odd fields displayed first and the even fields displayed second. Together, both fields create a full frame, made up of all 1,080 pixel rows or lines, every 30th of a second.
In 1080p, each frame of video is sent or displayed progressively. This means that both the odd and even fields (all 1,080 pixel rows or pixel lines) that make up the full frame are displayed together. This results in a smoother looking image, with less motion artifacts and jagged edges.
Differences Within 1080p
1080p can also be displayed (Depending on the video processing used) as a 1080p/60 (Most common), 1080p/30, or in 1080p/24 formats.
1080p/60 is essentially the same frame repeated twice every 30th of a second. (Enhanced video frame rate.)
1080p/30 is the same frame displayed once every 30th of a second. (Standard live or recorded video frame rate.)
1080p/24 is the same frame displayed every 24th of a second (Standard motion picture film frame rate).
For more details on how video frames are processed and displayed on a TV, refer to my article: Video Frame Rate vs Screen Refresh Rate
The Key is in the Processing
1080p processing can be done at the source, such as on a Upscaling DVD Player, Blu-ray Disc Player, or HD-DVD player - or it can be done by the HDTV itself.
Depending on the actual video processors used, there may or may not be a difference in having the TV do the final processing (referred to as deinterlacing) step of converting 1080i to 1080p.
For instance, if the TV is utilizing a Faroudja, DVDO, Silicon Optix/IDT HQV, or homegrown processors, such as the ones used in higher-end Sony, Samsung, Sharp, Sony, and Panasonic sets for example, may be equal to the processors used in many source components - so the results displayed on screen should be equivalent, or very close. Any differences would be more noticeable on larger screen sizes.
1080p, Blu-ray Disc and HD-DVD
Also, keep in mind that with both Blu-ray and HD-DVD (although HD-DVD has been discontinued there are still players in use), the actual information on the disc itself is in the 1080p/24 format (Note: There are some instances of content being placed on a Blu-ray disc in either 720p/30 or 1080i/30, but those are exceptions, not the rule). Most Blu-ray Disc players have the ability to output 1080p/24 to a compatible TV in that native form. However, there are variations on how some players accomplish this task. Here are two interesting past examples from two players that are no longer in production, but are in still use.
The first example is the LG BH100 Blu-ray/HD-DVD combo player (no longer in production). Since not all HDTVs can display 1080p/24, when you connect the LG BH100 to an HDTV that does not have 1080p/24 input and display capability but only has 1080p/60/30 or 1080i input capability, the LG BH100 automatically sends its 1080p/24 signal from the disc to its own video processor which then outputs a 1080i/60 signal. In other words, this player can only output a 1080p signal if the TV is 1080p/24 compatible. This leaves the HDTV to do the final step of deinterlacing and displaying the incoming 1080i signal in 1080p.
Another example of 1080p processing, is the Samsung BD-P1000 Blu-ray Disc Player (no longer in production) - what it does is even more complicated. This Blu-ray player reads the 1080p/24 signal off the disc, then it actually reinterlaces the signal to 1080i, and then deinterlaces its own internally made 1080i signal in order to create a 1080p/60 signal for output to a 1080p input capable television. However, if it detects that the HDTV cannot input a 1080p signal, the Samsung BD-P1000 just takes its own internally created 1080i signal and passes that signal through to the HDTV, letting the HDTV do any additional processing.
Just as with the previous LG BH100 example. The final 1080p display format depends what deinterlacing processor is used by the HDTV for the final step. In fact, in the Samsung case, it may that a specific HDTV has better 1080i-to-1080p deinterlacer than Samsung has, in which case you may see a better result using the deinterlacer built into the HDTV.
1080p/60 and PC Sources
It is also important to note that when you connect a PC to an HDTV via DVI or HDMI, the graphic display signal of the PC may indeed be sending out 60 discreet frames every second (depending on source material), instead of repeating the same frame twice, as with film or video based material from DVD or Blu-ray Disc. In this case, no additional processing is required to "create" a 1080p/60 frame rate via conversion. Computer displays typically don't have a problem accepting this type of input signal directly - but some TVs might.
In the final analysis, the proof is in the actual viewing - how the image looks to you in the real world with your specific HDTV, in combination with your source devices. Short of having a tech come out and doing actual measurements, or comparing results using different TVs and source components yourself, even if you don't have a 1080p input capable Television, as long your HDTV has 1080p internal processing, you may still be able to get the benefits of 1080p. The key is in the processing, and, of course, not all HDTVs and video processors are created equal - let your eyes be your guide.
For more on 1080p, check out my additional reference article: 1080p Facts and an article offering useful tips when buying an HDTV at About.com's site for TV/Video: Which HDTV Should You Buy: 720p, 1080i, 1080p?.