SmartThings supports two mesh protocols: Zwave and zigbee (using the Zigbee Home Automation profile, ZHA).
In a mesh network, many devices can pass along a message intended for another. This allows you to extend the range of your home network beyond the range of a single device.
The exact range varies depending on the local architecture, in a typical residential home in the United States we will get about 40 feet of signal indoors from each Zigbee or classic ZWave device. A Zwave Plus device will have a somewhat longer range. However, just as with Wi-Fi, there may be dead spots in the home for any protocol due to large metal objects including cars in a garage, concrete, types of insulation, water pipes in the wall, tinted double glaze Windows, brick, etc. In addition, boosted WiFi can interfere with Zigbee, reducing signal transmission.
Begin by placing the SmartThings hub a minimum of 3 meters away from the WiFi router. Then add repeaters as needed to get through dead zones and extend the signal throughout the home.
Battery-powered devices do not act as repeaters.
Most mains-powered devices that have a Zwave or Zigbee radio will act as repeaters. The exception is security alert devices like smoke alarms which need to remain focused on their primary task and so do not repeat for other devices.
"Auxiliary" or "Add on" light switches that do not have a radio cannot repeat even if hard wired.
Zwave devices only repeat for other zwave devices using the identical Zwave frequency. Zwave plus repeaters have a significantly longer range than classic Zwave. Both Zwave Plus and Zwave Classic Devices are compatible with SmartThings, and a repeating device can be of either generation. However, a device on the US frequency cannot be a repeater for the device on the European frequency or vice versa. Z wave is limited to a maximum of four hops per message.
Zigbee devices only repeat for other zigbee devices of the same profile. SmartThings uses the Zigbee Home Automation protocol (ZHA 1.2). Devices using other Zigbee protocols like Zigbee Pro Or Zigbee Green Power are incompatible with SmartThings and cannot be used as repeaters with SmartThings. Zigbee home automation allows for up to 15 hops into the hub and 15 hops out from the hub for a total of 30 hops to carry one message. So even though each individual hop is shorter with zigbee, the total area covered may be larger.
Bulbs of any brand attached to a Phillips hue bridge will only repeat for other devices attached to that specific bridge. These use the Zigbee Light Link (ZLL) standard.
Zwave locks require that the Zwave repeater closest to them support "beaming."
To determine if a particular Zwave device supports beaming, review its "conformance statement" at the official Zwave Alliance Products site.  Note that this is an optional feature, so it can vary even from model to model in the same line.
Prior to 2015, people used to buy devices that were dedicated repeaters. With the latest generations of Zwave and Zigbee this is no longer necessary, and you can use a double duty device which has its own primary purpose but will also repeat messages for other devices on the network.. In the SmartThings community, popular repeaters include plug-in appliance modules ("pocket sockets"), wired receptacles, wired light switches, and those lightbulbs which are able to repeat for nonlightbulbs.
As of October 25, 2015, the following devices in the US are examples of repeaters for different use cases, although there are many other devices that can also work.
Zwave Plus with beaming:
DragonTech Appliance Module, PA100, can work with LEDs and CFLs as well as some small appliances, and has a grounded outlet.
DragonTech Dimmer Module, PD100, can also work with LEDs and CFLs. Note that these models do not have a second non-network to pass through outlet. Typical cost is around $30, but they are often sold in a bundle with a MiniMote for under $50.
GE ZW4104. Pocket socket with no networked passthrough. Typical cost around $45.
GE ZW4201. Outdoor-rated pocket socket. Single outlet connector. Typical cost $60.
GE ZW1001. Wired receptacle. One networked outlet, one nonnetworked.
GE 14294. Wired Dimmer Wall Switch.
GE 14314. Wired Fan Control Wall Switch.
TKB Home TZ68CH. Pocket Socket. Typical cost around £28.
TKB Home TZ69CH. Dimming Pocket Socket. Typical cost around £29.
TKB Home TZ69CH Pocket Socket with energy monitoring. Typical cost around £36.
Most in wall micros will also support beaming. If you need something that does not require a neutral wire to go behind a light switch, the Aeotec Nano Dimmer ZW111 will work. Typical cost around £51.
Zwave plug-in Motion Sensor
IKEA Tradfri pocketsocket. Typically costs just under $10. Available in both the US and the UK. This appears to be a well-made budget device, no special features, but works well as a zigbee repeater even for Xioami sensors. The form factor isn’t great because it’s a little big, so you can’t plug two of them into one wall receptacle and it itself only has one socket. But as a simple on/off device that also works very well as a zigbee repeater, it’s a good choice. (The handheld buttons and remotes have only partial integration with SmartThings, so don’t bother getting those.)
IKEA Tradfri bulbs also work well as zigbee repeaters as long as they are on power.
GE appliance module (pocket socket). Typically costs $40 to $50. Has additional nonnetworked pass through outlet. (Note: the Zigbee and zwave GE devices look nearly identical, choose carefully to get the protocol you need, especially if buying off the shelf at Best Buy.)
GE light switch model 45856GE. Typical cost $50 at Best Buy. (Select carefully, the zwave, zigbee, and Bluetooth versions of the GE Switch look identical, so check the box.)
SmartThings SmartPlug. Typical cost $25.
ZLL bulbs will repeat for other ZLL bulbs. Some community members have been able to get ZHA bulbs connected directly to the SmartThings hub to act as ZHA repeaters, but others have reported these have proven less reliable than other zigbee repeaters.
Note that the Sengled brand of zigbee bulbs do not repeat for anything else, not even other bulbs.
Zigbee repeater for 220/240v countries using an XBee board. Because there aren't a lot of ZHA devices available in the EU, UK customers have mostly had to rely on the SmartThings smart plug as a zigbee repeater, or since 2018, the IKEA devices. However, one community member in the UK was able to put one together using an XBee board and an aerial which allowed him to get signal out to a second location. The discussion thread is quite technical, but it does show it can be done. 
Both Zwave and Zigbee in one unit
A few devices contain both a Zigbee and a zwave radio in one case.
Lowe's Iris Smart Plug 3210-L is a Zigbee pocket socket which was intentionally created for multiprotocol systems and includes a zwave plus chip whose only purpose is to act as a repeater. So this device can repeat both Zigbee and zwave messages. It also supports beaming. Typically costs about $35. It will need to be paired twice to SmartThings, once as the zigbee and once as the zwave. You can control the pocket socket with the zigbee device.
Some people have reported that the Z wave repeater will only work when the zigbee switch is turned on, while others have reported that the Z wave repeater just doesn't work very well, and they've gone back to using it only as a zigbee pocket socket. There is a community-created device handler for those who want to try it.