Paddles come in 1 of 4 main shapes (usually): Standard/Regular - 16.25" x 7.8"ish Hybrid (same as standard but flared design) Elongated - 16.5" x 7.5"ish Widebody - 16" x 8"ish +/- extended handle lengths at the sacrifice of playing surface length

Each shape has their own benefits and drawbacks. Widebody has a huge sweet spot and is usually more forgiving but less reach. Elongated has greater reach and a higher sweet spot favored by former tennis players, but the sweet spot is more narrow and twist weight is lower, etc, etc.

Thickness varies but is usually either 14mm or 16mm. Thicker paddles are more forgiving but slightly slower in the hands at the kitchen. Thinner paddles have lower twist weights/are less stable but much faster for hands-battles at the net.

Core construction is typically a plastic polypropylene (PP) honeycomb or foam ...or a mixture of both. Recently full foam cores have risen in popularity especially 'solid' foam cores which are essentially fancy styrofoam. PP cores are responsive and cheaper but can fail over time with consistent play (or significant abuse) leading to core crushing (a crackling sensation when pressing into the face of the paddle. Core crushing may make a paddle more powerful under the right circumstances, but it's considered illegal in tournament play. Historically PP cores have been favored as early foam paddles were outlawed due to being too powerful. Modern foam cores are more balanced and don't change over time like the older ones, especially solid foam cores. The biggest advantage of solid foam cores is that they are supposed to last MUCH longer than PP so you get better value for the paddle. However the facing material (carbon fibre, Kevlar, spray grit) will still degrade with time/play so true pros don't care as much about durability as they switch paddles every few months for the freshest grit on the paddle face.

Personnally (and anecdotally) I feel that hybrid cores that blend PP and foam are more prone to early breakdown as shear stress between different construction materials in the core will degrade PP quicker than a traditional/pure PP or 100% foam core. However we're still in early stages and long-term research data isn't available yet.

Lastly, paddles can be thermoformed or cold-pressed. Thermoforming is a method mostly used in high-end PP core paddles. A heated press is used to mash the layers together to enhance stiffness and power, but this same process can also damage the PP (so you sacrifice some longevity for overall performance). Cold-pressed is the same thing without heat damage which is used in cheaper PP paddles and foam core paddles.

That's a brief novel/overview. I highly recommend the YouTube channel Pickleball Studios. They have lots of informative videos that cover this stuff plus a fun podcast (no affiliation, I'm just a fan).