How can the router choose between two routing updates of different protocols?

by compare between the administrative distance (ِAD),the AD known by default and can change it manual

If your router is running multiple routing protocols, Administrative Distance is used to determine which routing protocol to trust the most.
Lowest administrative distance wins If a router receives a RIP and an OSPF route to the same network, it will use Administrative Distance to determine which routing path to choose.
The Administrative Distance of common routing protocols (remember, lowest wins): External BGP 20 Internal EIGRP 90 IGRP 100 OSPF 110 IS-IS 115 RIP 120 External EIGRP 170 Internal BGP 200

Router sources assigned what's called "Administrative Distance" , it's a value given to different route sources according to the following table ( default value - which you can change while configuring if needed) : Protocol Administrative distance Directly connected route 0 Static route out an interface 1 Static route to next-hop address 1 EIGRP summary route 5 External BGP 20 Internal EIGRP 90 IGRP 100 OSPF 110 IS-IS 115 RIP 120 EGP 140 External EIGRP 170 Internal BGP 200 Unknown 255 The router will chose the route with the lowest administrative distance.
for example , if a router has 2 routes to network 10.1.1.0 from 2 sources ( whatever they are ) , it will chose the one with the lowest administrative distance.
Again , the above default administrative distance can be changed while configuration.
Router(config)#router rip Router(config-router)#dis Router(config-router)#distance ? Administrative distance Router(config)#router eigrp 6 Router(config-router)#distance eigrp ? Distance for internal routes Router#sho ip protocols Routing Protocol is "eigrp 6 " Outgoing update filter list for all interfaces is not set Incoming update filter list for all interfaces is not set Default networks flagged in outgoing updates Default networks accepted from incoming updates EIGRP metric weight K1=1, K2=0, K3=1, K4=0, K5=0 EIGRP maximum hopcount 100 EIGRP maximum metric variance 1 Redistributing: eigrp 6 Automatic network summarization is in effect Automatic address summarization: Maximum path: 4 Routing for Networks: Routing Information Sources: Gateway Distance Last Update Distance: internal 77 external 55

1st of all router looks for the more specific route then if both updates are the same then it'll looks for the AD 

If the router has more than one route for same destination , It will check longest prefix match and thereafter smaller AD value for the given two routing protocol.

A router choses routing updates from difference routing protocols by the administrative distance of the routing protocol. The one with the lowest administrative distance will be chosen.

Best Route is placed in Routing table of router.More Specific routes are first placed and then comes Administrative distance in below sequence:

Connected interface 0 Static route 1 Enhanced Interior Gateway Routing Protocol (EIGRP) summary route 5 External Border Gateway Protocol (BGP) 20 Internal EIGRP 90 IGRP 100 OSPF 110 Intermediate System-to-Intermediate System (IS-IS) 115 Routing Information Protocol (RIP) 120 Exterior Gateway Protocol (EGP) 140 On Demand Routing (ODR) 160 External EIGRP 170 Internal BGP 200 Unknown* 255

for multi Routing Protoclo enviroment Administrative Distance(AD) is used as a tie Breaker if a route recive two route with different metrics

1) RIP with hop count3( metric3)

2) EIGRP with metric1213555 

the route do not compare the two metrics since this metric are calculated with different algorism so the router choose the route with low Administartive distance which is EIGRP Route

look below AD of some Routing Protocol

Protocol                                      Administrative Distance

RIP                                                         120

OSPF                                                     100

EIGRP                                                    90

IS-IS                                                       115

        

by adminstrative distance and using routing table its update

OSPF because it is cost effective approach

If a router has two or more than two routes for same destination in routing table :

the first thing that will mater is : which one of them is the longest match,

2nd will be the lower adminitrative distance value assigned to routing protocol.

for your reference

Connected routes :0

static Routes :1

RIP :120

EIGRP :90

OSPF :110

IBGP:200

EBGP :20